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iEEE Transacrions on Power Sysæms, Vol. 10. No. 3, August 1995
A DES ENS I'I' IZED CON'I'ITO L Lt, IT IIO R VO LTA G E RI'G ULATION
1,16 1
OF POWER SYSTEN,IS
A. Heniche, H. Bourlès, Ilember. l[EE. and iVI. P. Hourv
Abstract:
In this paper a robust linear voltage contrullr:r is proposcd
to intprove power svstcln stabilitv and ltcrforrnaucc unclcr
varvlng operating cr_rnditious. This controllcr includes the
autonratic voltage rcgulutor and thc po\vcr svstcur stabilizer.
It lirnits the variations ot thc tcrrninal vrjltage, <Iaurps thc
electromcchanical crsciltatiou and iucrcases thc cxcitution
during and aftcr a short circuit iu all operatinq points.'fhc
rlisiqn nrethod choscn to irnltrovc the purartrctric robustlress
is desensjtivity. It is the tirst tinre that I puranrerric robust
design nrethod has bcen applir:rl to rcsohc the problcur ot
voltage regulation. 'fhc nrcthocl principlcs when rhc
paranleter ts a l'cctor. arc prssentcd.'l'ltc dcscnsitizsd
controller has a sirnplc structure and therelirrc can c:rsjly bc
iurplemcnted. The rcsuits obti.rincd rr ith a sirnulation sottryare
and rçith the Eiectricité de Iirance Trunsicnt Nr:tryork
Ânalvscr slrorv lhe goorj pe rtbrnr:rncc and the ucjvantages of
the corrtroller u'ith rcspccl lo vlrious disturbanccs.'l'his
nlethod can be applicd to all kinds of gctrcrators. and is
parliculârlv rvcll suitcd in thc casc ol'lurbtrgcncrators
working iu verv diffcrent operatiug couditions.
Kevword s : Desensirivirr'.lrrrlr-,",ri. r.o[rusrness. st:]l)rll t\..
voitùÈe t c g uhtot. po\r'er svstern s tliri lizr:r..
1-Introduction:
We plopose in rhis pepel u lltethod for dcsrrrrrrrr: r !ollt!e
conuoller. includinq I voitlqe rerulutor.(ÂVllt and ù powcr
systcrn stalrilizer (PSS). This pr.ohiem is consrdcred as I u'hoie.
hcnce the ohtained ,\VR and PSS ur.e coor.dinrrect. The_rjlsign
model rr'hich is consi<ieled is the cilssiuai sr.srour <slnrlÈ
rnachine. int'irrite hus".
A good voltaqe lcgulation is vcrv iurltor.tlnr for. netrr,oLk
stabiiity. and this topic has ireen reccivirrg a lor ".r Jtrcntiuu lur
manv vcrts. For al.l lchnissilrlc oper.utrng poults. Jn actron on the
excilaùon voltaqe has to.
- cnsul'e slstem stulriiiLr'.
- tiatnp thc elecuomechlnrcei oscrtlatiou.
- echievc voltJte teguiullolt wlth()ut stiltlc elt.or.
- plovrde tooii pcrtoun:inces in cusc of shorts slriurLs.
swrtchirrg oper:rt.ions unci [rus licquency vu.iutrons.
The lltcer vlduLions modcl ttrrc kinJ of irrter.rctrons t)otw,een the
machlne under considcratioll i.llld the othsr ones. For Jli ruuchinss
of the sJmc tvpe. \r'c tnuke thc sJllrc Iullli]g ol thù colttl.oijcr
Ilectlicité de Irrmce. Dilcction
1 .,\l'enuc tiu (iénérai dc Gaulle. tjes Etudes et. Recherches
92 l-l I Clunu-t. ITRANCE
par:.u1tcters. According to EclF experience. this very particuiar
nono-rnachine apploach is etTicient to soive the multi-machine
proDletn.
One ot the difficulties of this problem is due to ttre different
kinds of disculbauces to be lcjected. For exarnple an <o-loop.
$'hc|e o is thc lotor speed, ullows us to increase the excitation.
and then the transienr stairiiity is iurproved. But, this loop
iuclcuscs thc tcnninal voitage variations when a bus tiequency
vuliuIion occuls.'fhcrel'ore. thc controller has to realize a
comlll o ln I sc be tw'ec n the cies iled perlbrmances.
.\ni,.rthcl diificulty is to obtain good pertbrmances tbr all
possiirlc Opc|aring points. As a lnatter of f'act, the behavior ot an
al[e|nator connected to a network depends. arnong other things.
on its position irr this netwolk. on the operrting conditions rin
part.lculu.-ùn thc lerctlve porver flow and on the voltage map). on
thc nclrrork topology and the generation schedule. Therefore, a
r.oitage col)tlol.ler lnust not he sensitive to changes of operatrng
porrtts. These changes are consi<iered here as purametric
unceltlrin ties.
Valrous rnethods have been used to improve and desiqn
volt:irtc contlollels. In ruost of thern. stabilizing signais at.e added
to tirc c'onvenrional voltlge ioop (AVR) to improve osciilation
Juuprnr !-61. .\VR generaiiy is a pr.ogrr.trouai and integral (PI)
lcguiator'. Some of these methods ue bascd on .rnodem control
[he rrLv", e.r. pole placernenr [7]. Linear Quadratic (LQ) oprirnal
contlol [8]. nonlinear conuol [9-11], and IJ- theory [11].
LJnibltunltciv, thcse approaches are not wcll suited fbr designing
il lobust corrtl.o.lleI with Ie'spect to parunetric uncertainties. As a
rurattcl of l'i.lct. at this point. nonlinear coutrol rnethods do not
plovide lobustness gualilutees. On the other hand. all methods
rclutcd to II- thcoly (such as g,synrhesis [13]) are very
conselvatrve rn the case oI str.uctured and reai uncertaint.ies (such
ls paralletl'ic ollcs. as in our problem) [14].
A ,scnelûtor connected to an inlinite bus is a nonlinear
svstsrll. çhich can lre lineurizcd a|ound an operating point
tlefinccl bv specific vulues P*. V*. Q* and X* of the active
power. of the tclnrinal volt;rce. of the reactive power and of the
e.rternJi Lerctance. Ilence, the lineanzcd modci depends on the
vector of pll'auretel's g* = [P* V* q* a*1r' 0* can vary rn one
tlomrin D. cullcd "adrnissible paralnetr.ic dornain>, which
chlrlctelizcs ull possible operating poiltts. Some of the
cor)rponcl)ts ol 0* (e,speciuily X*) are unknown. Moreover. in
olclcl to olrtain I coustûnt. linear controller. it is assurned in this
prpel thut rhc whole vecror 0*e D is urrknown lwher.eas D is
knrt*,n). The vuriutions of e * irr D lrc the Diirarneurc
ullccf ta llt Ilc s.
l'hc lun of this pupcr is to proposc ii rnethod for designing e
lobust controllcl rvith respcct to thesc pal.alncttic unccrtaintrcs.
i\lolc specitically, this controller is linear.. tilne-invu.iaut,
indcpcndcnt oI 8. und for evcly 0 e D:
?5 'lli{ 113-8 FWRS À paper recommendeci e:rj epproved
bi' ihe iEEl Fover System En6ireerin3 J,:r:_tt,ee oi
ihe IiEn Pcwer Ingineering Soc:ety lcr ::eseniation
rt r-he I995 IEEE/pES l{inter ileeting, .-ên,.r.ar,J/ 29, -,o
lebruary 2,'t995, lJew Ycrk, ilY. l.januscr::-, subrci.,tei
jul;r 26, -1994; rade available t'or ;:rintt::g
,ianua:;r 11, loo<.
0S85-8950/95/S0J.0O o 1995 IEEE

l+62
-provides sufficient robustness tnugius ti.e. gain. phrse. und
delay rnu'gins I I l4] ;.
-achieves voltage reguiation wiùlout st{rtlc enor.
-rninirnizes the eifects of distulblnces on the rnachirte
tenninai voltage.
The design problem ol a robust conuollel rvith lespcct to
parametric uncertainties is the suhjcct of uruuy studies todav. In
spite of these efforts. onl.v a t'cw mùthods cxist. lnost of rvhich ue
concerneci wrth analysrs oniy'1see, c.g. thc Khantonov approach
[5] and the rnixed p - anall sis I l6] 1. Our rpproach is
.desensitivitv'. It is a dcsign lncthod Lrascd on the Lineûr
Quadratic Gaussian {LQG ) optiurul corrtloi. rvhele Ihe cluliI'::tic
in<iex to be mrnirnize,.i rs incleussd [)v thc si-r-culled
.sensit.ivitresr. The i:rttcl ch.rLJe lsr rz-c Lhd vJrrJtiuns ul thc svstcrn
variables when 0 valies rn D.'fire first developments of this
method rvere urade b_'- scverli ruthors (sce. e.g. [17]) lnd *ere
studied in tnore depth urd pcrtèctcd [rv de Lurrninut en<] Bcrovich
[18.19] in the case *'hele 0 is a scaiar'. This method is genelrlized
here in the case where e is u tinire-dinrcl)tionrl vcctor. Notc that
in the above-rnentioned robust contloi urethuds. e.9.. F-svnthcsis
and Khalrtonov-based applo.rches. thc rtai)ility is a pliolr
guaranteeci tbr all rnodel errors LrelonginS. to the set. of adntrssrble
ones. The price to pa)" is the conscrvation 1as *'c slirl rLrove)
and/or the hudge complcxrty uf courput.r[ions li0]. In thc
desensitivitv lnt:thod. the unknorvn Dlliuneter d is assurned Lo be
a stochastrc one. with a SJusslull distliirution and kno*'n lilst urtd
second morncnts. We cortsicicr the iJlter as dcsrqn pulaurclcls,
i.e.as additional degrees ol frccdolll loL irnplovrng Lobustncss.
The stability of the contloiled svstem for all vuiues (or'. in
practice. for a representative sot of va.lues) of e in D lnust be
checked a posteliori. This approach hus nvo ildvant.rscs:
accordinq to our experience. it is nut eor)scrvJtive: luoleovcl lt
does not prcsent prohllems of colupuhtional efficicncy.
The c1esensitizing contlol plovides a Soo(i solutiun furl our
problern. Nlor'eover.:rs will [.e shown in Fig.i. thc resulrint
controller consists in foul ioops in pulilel on P - P,,,. co. V und
V-V.. r"'hele P,,,. {o and V. llc the mcchlnical power'. thc rotor
speeci and the cornurund signal, r'esprctrvciy-. It c.rn be intclplctcd
iu an A!'ll suppieurentecj try u.lrSS usinr as stabiiiz-rrtl signeis the
difference P - P,,, and <o.
The paper is olganized as lbllows: Scction 2 is dcvoted o the
model studv. In the next sec!ion. dcsensirivity is ciescriired.
Section.l contains the resuits. lnd Section 5 inclucics tlte
conclusion. Non standard rnathematrcùl tools (such ls li'oncckcL
products of lnauices), rs woll as sorne calcullLrons, i!s plcscnted
in the appendix. A trrief overvicw oi LQC theoly is givcn ru
Sectron .i. l.
2-Model study:
A machine conrlcctcd to ûn inlioitc [rus thlough utt exte tnll
reactallce can ire dcsclibed hy e nottiittcll tttoclcl blseci ott l)utk's
equations [21]. Llndel thc usuai rssuluption (tloll saturatloll, attd
Ji dY
tlerivatrves I rnd; of the curlent artd flux ule negligible r'''ith
respect to (l)i und <oY). u fifth orcle| uoniitteu' rnodcl is olrtuincd
[22.|. As a l.ineal rnodel is neccssi!'.v to JcsiSll a lirteal cottttoilct.
the nonlineu' rnodel is Iincarizcd [r,v deviution tLotn lll upcrrLilr{
point. Bv using the urodcl teduction tnethod bùscd on the
balanced lellrzation [231. u classicrl thild otclcr'lirtcut tnotlcl
I Th",l"luu tuurgirt is rhe stullies! uttttrodcllcci .lcilv rvhrch
rnakes thc ciosecl loop unstattlc. l'his rnllgin is I *av lo tltkc irttr-t
accoullt unruoricileci .ivrlrntrcr i lJl.
which connccts the voltage excitation to the rotor speed. the
lcrrninal voltage and the active powcr is obrained. Finaliy, as the
linealize<j llioclel dePends on tbe vector of parameters 0' a
ruracbine counected to an infinite bus can be described by the
lbllowin g state equation:
x(l) = A(g)x(t) + B(0)u(t) + cr(t)
y(t) = x(t) 1= [rr: V P ]r x(OFxo (1)
rvhere ul.) is the state excitation noise. x0 is the initial state. u=Vf
rie excrration voltage and y the measured output; A(0) and B(0)
are statc lnd iuput murices. r'espectrveiy.
3- The desensitivity method:
As rvas said above, the difficultv of our problern principaily
lics in palarnctric uncertainties. \#'e have to ensure systeln
stabilitv as wcll as a good performance when tbe system
pluuuleLcrs vu'y. As opposed to the standard LQG method'
dcsensirivity includes pararneuic uncertainties at the design step
oi the rcquiator. The desensitizing control law minimizes the
clfects of puramcter valiations which can be interpreted in this
ccutext as distut'bunces to be rcjected.
3.1 A brief overview of LQG theory :
Consider l general stute equation of a linear model.
x(t) = Ax(t) + Bu(t) + cl.(t)
y(t) = Cx(t) + p(t) x(0)=xo
u'hele tr(.) and [J(.) ale the state excitation noise, and the
tne:lsuferncnt noise respectively. Both oi them are assumed to be
wirite anci gaussian. x6 is the initial state. u the input and y the
lulcasured output: A and B are state and input mauices,
lcspectrvelv. The classical problern LQG is to find the regulator
K which rninimizes the quacilatrc index
J= rirn I Tot*t,,, Q. x(t)+ur(t) Rc u(r)ldt
Q. is a nonnegative clefinite syrmnetric matrix which weights the
variations of the state around its nolnlnal value. R" is a postttve
.ictinite slmnrctric matrix which weights the energy of the
contloi. With such a criterion the obtained regulator realizes a
tr'ade-otf between thr: state variations and the energy of the
conrol signai.
Note th:it in this c:rse the pararnetdc uncertainties are not
considered. Flowever. the stâte equation ( l) depends on a random
vuii:ble 0. Consequently, it is not possible to directly applv a
chssicd LQG dcsign rnethod to this systern.
'l'he dcsensitivity is a design rnethod based on the LQG
theory. This method takes into account the parametric vanaÙons
at thc desi[n step of the controller. The next section describes the
rurcthod principles. ancl shows how. ttrrough apProximations, it is
;nssible to use tie LQG theory tbr a systetn which depends on an
unccrtilln par]meter.
3.1 l\Icthod principles:
Considel the strle cquation ( l) where 0 e R' is a gausstan
llnclour pararncter. assumed to be tirne invariant and unconelated
- --T
\\'itho.(t), 0 = E(e ),0= g -0 and!= E( 00 )ale the norninal
valuc of e. the ccntrecl pararnetric error and his covariance.
lespcctivciy. It is assutned that 6 and I9 are
unkuorvn. The profrlern considered in this paper
conuollct' K rvhich minunizes the quadlatic index
tTrT
Jr= liLn ; lItx'(t,e) Q" x(t.e )+u'(t.0) R. u(t.6)ldt
'l'J- 1 ()
known; 0 is
is to find the
(2)

'T --T-
J,= -lim ; Je 1x'q"x - u'R. u- {V
r --tæ
-T
(V ur-r\ I R.trV utlJt
g--t1
,xrrt&8Q.) tVri)+
(3)
Obviously the solution rvouid bc a stundu'd LQG contlol hw if 0
woul<l be known. Suppose thrtù i, smrll. so thar a first order
Taylor expansion of x ancl u aroundd is a good apploxirnation of
them. Using Kronecker products (see Appendix,) provides an
approximation J2 of J', which is independent of the randorn
variable 0:
1463
rvhclc V is a block rnatrix with Vll= Qcl Vtz= O;Vrr=Ot Vzt=O:
V lr =0. V::=lL8Q.;.r IoSH; 1' (I1iSRcXI'OHi);
V::=( IpSHi ) ilceRc)(Ip@G; );V32=(IpOG1)' (&8R"XIo8Hi);
v33= ( IpSGi)' (I+8R")(Io8Gi);
The desensitizing control law is designed by using tbe LQC
theory on the increased rnodel (9). The minimization of (10)
provides a Kirl conEoiler. Tbe iterations are repeated until
convergence in the graph topology sense [24]. As all stâtes of the
thild older model (l) are accessrble. Kois a state feedback LQ
contr oi.ler.
J.2.3 Thc dcscnsitizcd controllcr:
This mcttrod provided a desensitized controller whose block
diagram is given in Fig.l. The controller is defined by an
equation u(s) = - K(s) zG) (in the Laplace domain), where z=[(ù
V (P - Prr) 1V- Vc )l t.The signa.l P. is used to obtain a
Icccitorwlrd efièct.
I
Fig.l: Thc gencrator with tlre desensitized controller
The considered exciter is a brushless one which can be modelled
tbr the design ls a flrst order transfer function.
.l-Resulfs:
The desensitized controller provides sufficient stability
rnargins in the whole dornain D. It has been tested with Matlab
soitrvare. "vità Eurostag, a tirne simulation software for stabilitv
studies [2.5] and with the Eiectncité de France Transient Network
Analyser in diiferent cases of step in V. (the voltage serpoint), of
three phase fault. of switcbing operation and of
decreasing/increasing of the bus frequency. [n this section we wiil
illustrate the perlbrmances of the proposed voitage controller
thlouqb sorne sirnulation resuits. in comparison with a classical
PSS using P as stabilizing signal. This PSS is calculated for P=Pn,
tl=0.95Un Q= 0.42Qn and X= O.65Zt
For diftèrent operating conditions. Fig.2 and 3 sbow the case
of a -.57o step in V" and a variation of bus trequency (2OO mHz in
7..5 scc) rcspectively. As opposed to the classical PSS, it appars
that the variations of the terrninai voltage are approximatively the
sune for every operating point with the desensitized controller.
'fhis is an eftèct of the desensitivity. The bus frequency variations
crn be iuteryreted as a distul'bance which essentially acts on or. A
too lalge gain on o is not appropriate to reject this kind of
disturbances on V. However. one observes in Fig.3 that the
voltare valiations with the desensitized conûoller are of the same
older ot rnlgnitude or srnaller than with the classical PSS.
Fig.4 shows the case of a three phase t'ault. During and after
thc flull it is necessary to increase the excitation to improve the
transicnt stability. The system is stable witlr the two regulalors.
but the cllssical PSS generaæs a decrease in the excitation signal
.just alter ùe fi:ull contrary to the desensitized control law which
kecps thc cxcitrtion at is highest value after the flult (see Fig.4).
- ù2' J2' 'r
where z= z(t,Q),Y z = [_ --) (rlso wrirtenz ) is rhe
o 'à0t don ' e.
sensitivity of z with respect to the valiûtions of e, and I is the
Kronecker product.
In cornpanson with the classical LQG urethod, (3) shows that
in addition. tÉe desensitrziug contlol llw rnininrizes the smte and
controi sensitivities with respect to pJl'uureter variations. Ihis
minimization ailows one to irnprove the pu'uneuic lolrustuess.
We wiil show in what follows thrt sÈnsrlivitics cen Lre
calculaæd on the basis of the so-called "scnsitivity rnodcl> which
depends on the reguiator K to be cletcnnincd.
3.2.1 The sensitivitv nrodcl:
A LQG type conrroiiel cln lre describcd lry the state spirce
equations
i=ôE+fy u =- Gq- Hy (1)
Set o = i . t, = { lwirh the notation intlociuceci in section -1.2
u-g
and in Appendix). Differentiating ( l) with lcspect to 0 and using
the Kronecker product I'ieids (see Appendix ).
o = Aex + Bou + I"8(Ao+Bus) (5)
)'s = o (6)
Simiiary, dif ferentiating (.1) with respect to e f ic'lds
F=lr8(OF+l-yo)
ue=-ir8(GP+Hye)
]-'\
\t )
(8)
The sensitivitv model is chalacterized by o und p. Ir depends on
tle regulator K. which is to he designed. Thelctble, it is not
possible to mrnimize (4) in one step. urd rve rvill usc an iteratrve
method. Note that these itcrations arc ntade off-iine, so thùt the
gains of the rnstalled contloller do not chungc (i.c.. the cor)tlollcr'
is a non adaptatrve one).
3,2.2 ^lbe ltcratlve nrcthod:
Assurne that at stcp i. lhe regulator K1 is known. We rhcn
compute the sensrtivity rnodcl associlted to Ki. Conrbinitrg the
equations (5-8) and using the Kloneckel pr'oduct. the inclcascd
model used tbr the desiÈn rs:
0 0l
The quadratic inclex (3) can be wrttten
IIT_T
J:= lirn ; ig tXr'f'X ' + u' R-.u I Jr
'f+- r t'
IBl [*l
"=L:'l -.=L ; J
(e)
{t=A"Xr+Biu+o(t)
Y(t) = C, X"
[A 0 o I
A" = | Ao I.8(A-BFI1) I"A(-BGi) |
L o l"8fi Ip8or J
o'(t) = [c(r) 0 0 ]T Cu= [I
(10)

1.164
Fig.2: - 59lc ste p in V,
o=Pn L=r.95 Ll" . Q<1, Xl)..15 Z,r
P=Pn L:i1.95 L',r eil Xi).lU Zrl
P=Pr Ur).95 Ur Q=. Qn X=0.1() Zlr
P=0.5Po Lii.95 L'r Q=Qn ;{il.l0 4r
FiC.J: I l0 nls three ohase fault
P=Po U=l.u5Uu Q=Qn' X=O.45ZI
sr,tiil - ,
dÀ\lid({ - -,
LLluitctl - - -,
\.lolled .... j
o.g:-- - -- rh -:
\'..
c qç - - - - - - -\i
0
t.E
0.7
06
l.l I
u
f)...'rritiz..rl, "ntr nl l,'r'
lry
î
È ,.qF
tttF
o t6F
,18
{
. :.. . ..:
-----\\..
,*ts \::.
\r "
\
'lq?F \
lttr.-
ùE6ts -
Clissrc:rl I)SS using l)
Fig.3: Variation of bus frequcncy 1200 nrHz in 7.5s)
P=Pn ['=tir Q=ù X=ri.l0 Zll rsolirJ -
P=Pn tl=l:l Q= r).5 Qn X=0..1() Zil (.Ji6lld(,t - - r
P-Pn U=tlr Q=Qn XrJ.7t) Z,t rrlsircd - . - r
P=Pn Li=ti" Q<ln Xa].95 Z,r rdi,rred .... 7
ù.e8-
I q6ts
UûÈ (R'
Cllssicll I)SS Lrsirls l)
Classicsl PSS using P
PSS using P - Desensitized controller
I)SS using P - Desensitized controller (*) - . -(Zoom)
t
I
192
r') lhc vclticul linc indicutcs the cncl of thc fault

1465
One can observe a loss of stability with thc ciassicul PSS when
i-he fault occurs in a more coustlurned situation (see Fig.5). This
conf-rnns the need for an <o loop to irnprove ttansicnt stability.
Fig.6 shows that the behavior ol the desensiùzcd contloller in
the case of switchinrl opel'xtion is llso sutistactoly. The volr;rge
variations are smailer with the dcsensiùzcd conuollcr th:rrt *'ith
the classicai PSS.
Fie.5: ll0 tns tlrrec phasc fault
P=Pn t'=l.Jst'n Q=Qir x=,, ôzrr
ùJ l lJ I l.-r j
unE (Ê)
Classical PSS using P (Zoonr)
1.6 LE
PSS rrcinc P -. (Zoorn)
P=P" U=l.05Un Q=Qn X=(0.30+0.55)Zo (solid - )
p=pn t'=l.05Un Q=Qn X=(0.45+0.70)Zn (dâshdot - ' -)
P=P,r ['=l.u5Un Q{.5Qn x=10 40-0.701Z,n (dashed - - -,
p=pr t'=i,u5Un Q=0 X=(0.3ù+0.55)Zo (doued .... )
Fig.6: Switching opcration
9
b
I
,i
5-Conclusion:
'fhe results o[rtainccl ancl the structure of the controller shoù
that the desensitivity rnethod is well suited to soive the problem
of power systems voltage regulation. This controller can be
intelpleted as a coordinated AVR/PSS. like [26]. The linear
quudrutic stxte feedblck conuol used here allows one, through the
o loop. to iucrease the excitation after a three phase tbult and then
to iurplove transient stability. Moteover this loop does not result
in signiriclnt variltions of the voltage in the case of bus
tiequency valiltions. Desensitivity ûnproves robustness witb
rcspcct to paralnetlic uncertainties. The desensitized controller
keeps the system stable in the whole admissible domain. Under
small riisturbances. tbe performances of the desensitized
controller a.re aknost independent of the operating conditions.
This is the consequence of the minimization of state variations
whcn the systeln paralneters vary. The results regarding this
voltage contr'oller behavior on a multi-tnachine network will be
givcn elsewhele.
Acknolvlegdements:
The authols would like to acknowledge for his assistance P.
.lustou rvho studicri some practical aspects of this piece of work.
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Appendix:
The l(r'ouecker
Rnrxn andBe
L =AW
Let NI(0) and
Jcpent-ttnt on
(/W' dW'
,rt t -\
v1e ) be a mxn mauix and nxl vector, bortr of them
a veclor 0 e RP, and set w=Mv, w, = Vrw = [
rf rr _ r aNt....rtvl rr
;' , .\r . = t:;-'. ::- 1. . Differentiaring w wirh
u c/u I dv^
ploduct C e RmrxnPof two matricesAe
RrxP,isdetinedby:
Ias18... alnBl
B= | |
La-18 ... an,nBJ
le spect to 0 vields w,, = lvl,, v + ( In I M; v. where [o is the
0{J
rientitv lnrtrix of tlirucnsion p.
Biographies:
.-\nnissa Heniche was boln at Bordj Menaël (ALGERIA) in
1962. She has adegree in Engineering fiom the " Ecole Nationaie
des In[enicurs D'Algerie" (,i985) and a master in automatic
coutrol fiorn "t-lniversité Paris ll" (1992). She is currently
prcoaled l Ph.D on robust conuol and his application ro power
svs[culs volLrsc legulation at the Electr.icité de France Researcb
Ccntre.
l)fi Ilerrri Bourlès was lrorn :rr Quimper (FRANCE) in t9.54 and
has a tjeu'ce in Engineering trom the ''Ecole Centr.ale". Pans,
Frrnce. IIe obtained his Ph.D in 1982. I.lis rhesis, prepared under
rhe suidance ot Ph'Ioan Landau. was on robustness theory. In
1992. he obtained his "habilitation" to fill a universrrv char.
Florn 1982 to t9ti7, he was a Professor in Mathemarrcs. signai
plocessiurl ilnd tulolnatic control at the "Ecole Supérieure
,1'Inforrnatique d'Electlonique et d'Autornarique" ( ESIEi\ ), Paris.
Fr':rnce.
Since l9tl6. hc has been rvolking as a Research Engineer at the
Elecuicité clc France Research Centre. He has also heen terchinq
unclelg|aduate und graduatc studeuts at university. I'lis marn flelds
.rf rcscurch llc: robustness theorv. system theory and applicaûon
ui autorna[ic contlol to power systcms regulltion.
\'luie-Pierle llouly was lrorn at Dunkerque (FRANCE) in 1968.
She hrs a deqree in Engineering from the "Ecole Supérieure
tl"Electlrcité" ( 199 I ). She joinecl rhe Electriciré de France
Il.cseu'ch ('cntle in i991. She is presently a member of the team
worklnq on power systelns dynarnics and control.