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Generators
G. Damm, Riccardo Marino, Françoise Lamnabhi-Lagarrigue
To cite this version:
G. Damm, Riccardo Marino, Françoise Lamnabhi-Lagarrigue. Adaptive Nonlinear Excitation Control
of Synhronous Generators. Nonlinear and Adaptive Control, 2002, 978-3-540-45802-9. �hal-02936801�
GilneyDamm 1
, RiardoMarino 2
,andFranoiseLamnabhi-Lagarrigue 1
1
LaboratoiredesSignauxetSystemes,CNRS
Supele,3,rueJoliot-Curie
91192 Gif-sur-YvetteCedex,Frane
dammlss.supele.fr, lamnabhilss.supele.fr
2
DipartimentodiIngegneriaElettronia,UniversitadiRomaTorVergata,
viadiTorVergata110
00133 Rome,Italymarinoing.uniroma2.it
Abstrat. In this paper, ontinuing the line of our previous works, anonlinear
adaptive exitation ontrol is designed for a synhronous generator modeled by
a standard third order model on the basis of the physially available measure-
ments of relative angular speed, ative eletri powerand terminal voltage. The
powerangle,whihisaruialvariableforthe exitationontrol,is notassumed
tobeavailableforfeedbak,asmehanialpowerisalsoonsideredasanunknown
variable. The feedbak ontrol is supposed to ahieve transient stabilization and
voltageregulationwhenfaults ourtotheturbinessothatthemehanialpower
maypermanentlytakeany(unknown)valuewithinitsphysialbounds.Transient
stabilizationandvoltageregulationareahievedbyanonlinearadaptiveontroller,
whihgeneratesbothon-lineonvergingestimates ofthemehanialpowerand a
trajetorytobefollowedbythepoweranglethatonvergestothenewequilibrium
pointompatiblewiththerequiredterminalvoltage.Themainontributionshere,
omparedwithourprevious works,istheuseofon-lineomputationandtraking
ofequilibriumpowerangle,andtheproofofexponentialstabilityofthelosedloop
system for states and parameterestimates, instead of the previous asymptotial
one.
1 Introdution
Theproblemofstabilizationofpowergeneratorsisalassialpowersystems
and ontrolsystemsproblem. It hasbeenapproahedforsometime nowin
manyworksinitiallybylassiontrolandlinearmodernontroltehniques
with good results, but only loally valid. Reently this problem has been
treatedbynonlinearmethods asLyapunovtehniques(seeforinstane[12℄,
[10℄,[7℄).
Reently, feedbaklinearization tehniques wereproposed in [6℄,[4℄ and
[13℄ to design stabilizing ontrols with the purpose of enlarging the stabil-
ity region of the operating ondition. Nonlinear adaptive ontrols are also
proposedin[1℄and[14℄.
Thenonlinearfeedbakontrolalgorithmssofarproposedintheliterature
make use of power angle and mehanial power measurements whih are
physially not available and have the diÆulty of determining the faulted
equilibrium value whih is ompatible with the required terminal voltage
onethefault(mehanialoreletrialfailure)hasourred.
Followingthelines of ourpreviousworks [5℄and [3℄wemakeuseof the
standardthird order model used in [14℄ (see [2℄ and [15℄) to show that the
terminal voltage, the relative angular speed and the ative eletri power
(whihareatuallymeasurableandavailableforfeedbak)arestatevariables
in thephysialregionofthestatespae
Inthispaper,ontinuingourpreviousresults,westudythezerodynamis
ofthesystem,withrespettotheterminalvoltage,fortypialvaluestoshow
the existene of two equilibrium points, one stable and oneunstable. This
is a motivation for the useof nonlinear ontrol instead of the lassialone
omputedusingtheapproximatelinearized modelaroundthestable point.
In our previous work [5℄, a nonlinear adaptive feedbak ontrol on the
basis of physially available measurements (relative angular speed, ative
eletri power and terminal voltage) waspresented. There, when apertur-
bationourred,thesystemwasmaintainedin theoldequilibriumpoint,no
longer valid, ausing wrong outputs, while the estimation of the new equi-
librium point wasmade.Then, atrajetory todrivethe systemto thenew
equilibrium pointwasomputed.
Inthe presentwork,estimation of thenew equilibrium point and om-
putation of the trajetory that drives the system there are done on-line.
Global exponentialstability isguaranteed forthe whole losed loopsystem
to this new previouslyunknown equilibrium point. There is a onsiderable
improvement with respet to the output errorswith the on-line proedure,
androbustnessisguaranteedbytheexponentialstability.
2 Dynamial model
Asin [5℄,weonsiderthesimplied mehanialmodelexpressedin perunit
as
_
Æ=!
_
!= D
H
!+
!
s
H (P
m P
e
) (1)
where:Æ(rad) isthepowerangleofthegeneratorrelativeto theangleofthe
innitebusrotatingatsynhronousspeed!
s
;!(rad/s)istheangularspeed
of thegeneratorrelativeto thesynhronousspeed!
s
i.e.! =!
g
!
s with
!
g
beingthegeneratorangularspeed;H(s)istheperunit inertiaonstant;
D(p:u:)istheperunitdampingonstant;P
m
(p:u:)istheperunitmehanial
input power; P
e
(p:u:) is theperunit ativeeletripowerdeliveredby the
generatorto theinnitebus.Notethattheexpression! 2
=!
g
issimpliedas
! 2
s
=!
g '!
s
intheright-handsideof(1).Theativeandreativepowersare
givenby
P
e
= V
s E
q
X
ds
sin(Æ) (2)
Q= V
s
X
ds E
q os(Æ)
V 2
s
X
ds
(3)
where: E
q
(p:u:) isthequadrature'sEMF;V
s
(p:u:)is thevoltageat thein-
nitebus;X
ds
=X
T +
1
2 X
L +X
d
(p:u:)isthetotalreatanewhihtakesinto
aount X
d
(p:u:), the generatordiret axis reatane,X
L
(p:u:), the trans-
mission line reatane,andX
T
(p:u:), thereataneof thetransformer,and
thedenitionX
S 4
=X
T +
1
2 X
L
.ThequadratureEMF,E
q
,andthetransient
quadratureEMF,E 0
q
,arerelatedby
E
q
= X
ds
X 0
ds E
0
q X
d X
0
d
X 0
ds V
s
os(Æ) (4)
whilethedynamisofE 0
q
aregivenby
dE 0
q
dt
= 1
T
d0 (K
u
f E
q
) (5)
inwhih:X 0
ds
=X
T +
1
2 X
L +X
0
d
(p:u:)withX 0
d
denotingthegeneratordiret
axistransientreatane;u
f
(p:u:)isthe inputtothe (SCR)amplierof the
generator;K
isthegainoftheexitationamplier;T
d0
(s)isthediretaxis
shortiruittimeonstant.
Espeially beauseP
e
is measurable while E 0
q
is not, it is onvenientto
expressthestatespaemodelusing(Æ;!;P
e
)asstates,whihareequivalent
states aslongas the powerangle Æ remains in the openset 0<Æ <, as
follows.
_
Æ=!
_
!= D
H
!
!
s
H (P
e P
m )
_
P
e
= 1
T 0
d0 P
e +
1
T 0
d0
V
s
X
ds
sin(Æ)[K
u
f +T
0
d0 (X
d X
0
d )
V
s
X 0
ds
!sin(Æ)℄
+ T 0
d0 P
e
!ot(Æ)g
(6)
in whih(Æ;!;P )isthestateandu istheontrolinput.
Note that when Æ is near 0 or near the eet of the input u
f
on the
overalldynamisisgreatlyredued.Notealsothat herewehaveintrodued
thenotation
T 0
d0
= X
0
ds
X
ds T
d0
Thegeneratorterminalvoltagemodulusisgivenby
V
t
=
X 2
s P
2
e
V 2
s sin
2
(Æ) +
X 2
d V
2
s
X 2
ds +
2X
s X
d
X
ds P
e ot(Æ)
1
2
whihistheoutputofthesystemtoberegulatedtoitsreferenevalueV
tr
=
1(p:u:)
2.1 PowerAngle
Thepowerangleis notmeasurableandis alsonotaphysialvariable tobe
regulated;theonly physialvariable toberegulatedis theoutputV
t , while
(V
t
;!;P
e
)aremeasuredandareavailableforfeedbakation.
Asamatteroffat(V
t
;!;P
e
)isanequivalentstateforthemodel(6)(as
provedin[5℄)
Æ=arotg V
s
X
s P
e X
d V
s
X
ds +
s
V 2
t X
2
s
V 2
s P
2
e
!!
(7)
Iftheparameters(V
s
;X
s
;X
d
;X
ds
)areknown,statemeasurementsareavail-
able.From(7)itfollowsthat inorderto regulatetheterminalvoltageV
t to
itsreferenevalue(V
tr
=1(p:u:))Æshould beregulatedto
Æ
s
=arotg V
s
X
s P
m X
d V
s
X
ds +
s
V 2
tr X
2
s
V 2
s P
2
m
!!
(8)
Fromaphysial viewpointthenaturalhoieof statevariables is(V
t
;!;P
e )
whiharemeasurable.Thestatefeedbakontroltaskistomakethestability
regionofthestableequilibriumpoint(V
tr
;0;P
m
)aslargeaspossible.Infat
the parameter P
m
may abruptly hange to an unknown faulted value P
mf
due to turbine failures sothat (V
tr
;0;P
m
)may notbelongto the region of
attrationof thefaultedequilibrium point (V
tr
;0;P
mf
).The statefeedbak
ontrol should be designsothat typialturbine failuresdo notauseinsta-
bilitiesandonsequentlylossofsynhronismandinabilitytoahievevoltage
regulation.
AredutionfromP
m to(P
m )
f
ofthemehanialpowergeneratedbythe
turbine, hangestheoperatingondition:thenewoperatingondition(Æ
s )
f
isthesolutionof
(P
m )
f
P +
sin(Æ)
f
sin(Æ )
=0
andsine(P
m )
f
istypiallyunknown,theorrespondingnewstableoperating
ondition(Æ
s )
f
isalsounknown.
2.2 Zero Dynamis
If weregulatethe voltage output (V
t
) to itsreferene value (V
tr
), the zero
dynamiswillbegivenby
_
Æ=!
_
!= D
H
!+
!
s
H (P
m +
X
d
X
s X
ds V
2
s
sin(Æ)os(Æ)
V
s
X
s sin(Æ)
s
V 2
tr X
2
d
X 2
ds V
2
s sin
2
(Æ)
whihareveryomplex,andforsomeinitialonditionsorparametersvalues
maybeomeunstable.
Ifweuse thevaluesdened in[5℄ wemayplot!_ asafuntion of! and
Æ, therewill thenbetwopointsofÆthat satisfytheequilibrium ofthezero
dynamis.These points(thetworealones) areÆ=1:26andÆ=2:96.
Linearizingthesystemaroundeahoneofthesetwopoints,wewillhave
aseigenvaluesthepairs[ 0:31 7:58I; 0:31+7:58I℄and[ 13:86;+13:86℄
respetively.
Thus we have shown the existene of two equilibrium points, one
stableandoneunstable.Therewillthenbeanattrationregionforthestable
one. Ifoneis driven outof this region (by initial onditionsor byafault),
the ontroller will not at regulating ! and Æ and the system will beome
unstable. This shows that using the output error voltage asthe only error
signalmaybedangerousasonemayregulatethisvoltageandloosestability.
3 Adaptive Controller and Main Result
Inthissetionwepresentthealulationoftheadaptiveontrollerasin [5℄.
Ourmainresultisthentoprovetheglobalexponentialstabilityofthewhole
system,withparameterexponentialonvergene.
The model (6) is rewritten substituting P
m
by (t) whih is apossibly
time-varying disturbane:this parameterisassumed to beunknown andto
belongtotheompatset[
m
;
M
℄:wherethelowerandupperbounds
m
;
M
areknown.
Letf(;x) beaC 3
referenesignaltobetraked.Dene (
1
>0)
~
Æ(t)=Æ(t) f(;x)
!
=
1
~
Æ+ _
f(;x)
~
!=! !
=!+
~
Æ _
f(;x)
sothatthersttwoequationsin(6)arerewrittenas
_
~
Æ=
1
~
Æ+!~
_
~
!= D
H
!+
!
s
H
((t) P
e
)
2
1
~
Æ+
1
~
!
f(;x)
Dene(
2
>0;k>0) thereferenesignalforP
e as
P
e
= H
!
s
D
H
!
2
1
~
Æ+
1
~
!
f(;x)+
2
~
!+
~
Æ+ 1
4 k
!
s
H
2
~
!
+
^
while
^
is an estimate of = P
m and
~
P
e
= P
e P
e
so that (6) may be
rewrittenas(
~
=
^
)
_
~
Æ=
1
~
Æ+!~
_
~
!=
~
Æ
2
~
!
!
s
H
~
P
e k
4
!
s
H
2
~
!+
!
s
H
~
_
~
P
e
= 1
T 0
d0 P
e +
V
s
X
ds T
0
d0
sin(Æ)K
u
f +
(X
d X
0
d )V
2
s
X
ds X
0
ds
!sin 2
(Æ)+P
e
!ot(Æ)
H
!
s
2
1
+1+
1 D
H
(
1
~
Æ+!)~
+
D
H +
1 +
2 +
k
4
!
s
H
2
D
H
!
2
1
~
Æ+
1
~
!
!
s
H P
e
f(;x)
D
H +
1 +
2 +
k
4
!
s
H
2
^
_
^
D
H +
1 +
2 +
k
4
!
s
H
2
~
+ D
!
s
f(;x)+ H
!
s _
f(;x)
Dening
3
>0,wethenproposetheontrollaw
u
f
= T
0
d0 X
ds
V
s K
sin(Æ)
0
0
= 1
T 0
d0 P
e (X
d X
0
d )
X
ds X
0
ds V
2
s
!sin 2
(Æ) P
e
!ot(Æ)
+ H
!
s
2
1
+1+
1 D
H
(
1
~
Æ+!)~
+
D
H +
1 +
2 +
k
4
!
s
H
2
D
H
!
2
1
~
Æ+
1
~
!
!
s
H P
e
f(;x)
+
D
H +
1 +
2 +
k
4
!
s
H
2
^
+ _
^
k
4
D
H +
1 +
2 +
k
4
!
s
H
2
2
~
P
e D
!
f(;x) H
! _
f(;x)
3
~
P
e +
!
s
H
~
!
then,thelosedloopsystembeomes
_
~
Æ=
1
~
Æ+!~
_
~
!=
~
Æ
2
~
!
!
s
H
~
P
e k
4
!
s
H
2
~
!+
!
s
H
~
_
~
P
e
=
!
s
H
~
!
3
~
P
e
D
H +
1 +
2 +
k
4
!
s
H
2
~
k
4
D
H +
1 +
2 +
k
4
!
s
H
2
2
~
P
e
(9)
Theadaptationlawis( isapositiveadaptationgain)
_
^
=Proj
~
P
e
D
H
1
2 k
4
!
s
H
2
+!~
!
s
H
;
^
(10)
whereProj(y;
^
)isthesmoothprojetionalgorithm introduedin [11℄
Proj(y;
^
)=y; ifp(
^
)0
Proj(y;
^
)=y; ifp(
^
)0 and hgradp(
^
);yi0
Proj(y;
^
)=[1 p(
^
)jgradp(
^
)j℄; otherwise
(11)
with
p()= (
M+m
2 )
2
(
M m
2 )
2
+2(
M m
2 )
foranarbitrarypositiveonstantwhih guaranteesin partiularthat:
i)
m
^
(t)
M +
ii) jProj(y;
^
)jjyj
iii)(
^
)Proj(y;
^
)(
^
)y
Considerthefuntion
W = 1
2 (
~
Æ 2
+!~ 2
+
~
P
e 2
) (12)
whosetimederivative,aordingto(9), is
_
W =
1
~
Æ 2
2
~
! 2
3
~
P
e 2
+!~
!
s
H
~
k
4
!
s
H
2
~
! 2
D
+
1 +
2 +
k
!
s
2
~
~
P
e k
D
+
1 +
2 +
k
!
s
2
2
~
P
e 2
Completingthesquares, weobtaintheinequality
_
W
1
~
Æ 2
2
~
! 2
3
~
P
e 2
+ 2
k
~
2
(13)
whihguaranteesarbitraryL
1
robustnessfromtheparametererror
~
tothe
trakingerrors
~
Æ;!;~
~
P
e .
The projetion algorithms (11) guarantee that
~
is bounded, and, by
virtueof(12)and(13),that
~
Æ,!~and
~
P
e
arebounded.Therefore, _
^
isbounded.
Integrating(13),wehaveforeverytt
0 0
Z
t
t0 (
1
~
Æ 2
+
2
~
! 2
+
3
~
P
e 2
)d+ 2
k Z
t
t0
~
2
d W(t) W(t
0 )
SineW(t)0and,byvirtueoftheprojetionalgorithm(11),
~
(t)
M
m +
itfollowsthat
Z
t
t0 (
1
~
Æ 2
+
2
~
! 2
)d W(t
0 )+
2
k (
M
m +)
2
(t t
0 )
whih, if W(t
0
) = 0 (i.e. t
0
is a time before the ourrene of the fault),
impliesarbitraryL
2
attenuation(byafatork)oftheerrors
~
Æand!~aused
bythefault.Toanalyzetheasymptotibehavioroftheadaptiveontrol,we
onsiderthefuntion
V = 1
2 (
~
Æ 2
+!~ 2
+
~
P
e 2
)+ 1
2 1
~
2
The projetion estimation algorithm (11) is designed so that the time
derivativeofVsatises
_
V
1
~
Æ 2
2
~
! 2
3
~
P
e 2
(14)
Integrating(14),wehave
lim
t!1 Z
t
t0 (
1
~
Æ 2
+
2
~
! 2
+
3
~
P
e 2
)d V(0) V(1)<1
Fromtheboundednessof _
~
Æ;
_
~
!and _
~
P
e
,andBarbalat'sLemma(see[9℄,[8℄)
itfollowsthat
lim
t!1
2
4
~
Æ(t)
~
!(t)
~
P (t) 3
5
=0
Wemaynowrewritethelosedloopsystemfollowingthenormalform:
_
~
x=A~x+ T
~
_
~
= x~
whihleadsto:
_
~ x=
2
4
1
1 0
1 (
2 +
2 )
!s
H
0
!
s
H
3 +
k
4
2
1
3
5
~ x+
2
4 0
!
s
H
1 3
5~
(15)
_
~
=
0
!
s
H
1
~ x
where
1 and
2 ,as
3
onnextequation,areonstants.Andthenomputing:
T
=
! 2
s
H 2
+ 2
1
=
3
>0
We then may show by persisteny of exitation that x~ and
~
will be
globally (for the model validity region) exponentially stable, then all error
signals go exponentially to zero, for all C 3
f(;x). This is also valid then
forthepartiularasewhere f(;x)=Æ
r
whereÆ
r
isgivenbyequation(8).
But, sine Æ
r
is aone-to-one smooth funtion of , it will onverge to the
orretequilibrium valueÆ
s as
~
onvergesto0,i.e.thereferenetrajetory
willonvergetotheunknownequilibriumpointandthenlim
t!1 (Æ Æ
s )=0.
4 Simulation results
Inthissetionsomesimulationresultsaregivenwithreferenetotheeight-
mahine powersystemnetworkreportedin [4℄withthefollowingdata:
!
s
=314:159rad/s D=5p.u. H =8s
T
d0
=6:9s K
=1 X
d
=1:863p.u.
X 0
d
=0:257p.u. X
T
=0:127p.u.X
L
=0:4853p.u.
Theoperatingpointis Æ
s
=72 o
, P
m
=0:9 p.u.,!
0
=0to whih orre-
spondsV
t
=1p.u.,withV
s
=1p.u..
It was onsidered a fast redution of the mehanial input power, and
simulatedaordingto thefollowingsequenes
1. Thesystemisinpre-faultedstate.
2. Att=0:5sthemehanialinputpowerbeginstoderease.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 40
50 60 70 80
( 1 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.4 0.6 0.8 1
( 2 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.4 0.6 0.8 1
( 3 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−0.4
−0.2 0 0.2 0.4
( 1 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.94 0.96 0.98 1
( 2 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1 0 1 2 3
( 3 )
(a) (b)
Fig.1.a1)RealÆ(-),CalulatedÆ(-.),Æ
r
(--) a2)P
m (-),
^
(--) a3)P
m (-),P
e
(--)b1)! b2)Vt b3)Controlsignal
Thesimulationswerearriedoutusingasontrolparameters
i
=20 1i3
=1 k=0:1
Fig.1.a1)showsthatthealulatedpoweranglemathesperfetlythereal
one.Italsoshowsthatthetrajetoryforthepowerangle(Æ
r
)goessmoothly
toitsnalvalue,andthat Æfollowsitperfetly.
InFig. 1.a2) one may see that the estimation of the mehanial power
is aurate, it may be veryfast ifwehange the parameters,and speially
iflargererrorsareaeptedforthestateandoutputvariables. Thismaybe
understoodbylookingat equation(10).
The eletrial power is also orretly driven to the mehanial one as
wesee in Fig. 1.a3).Thesame may be observedin Fig.1.b1) for therotor
veloity.
Fig.1.b2)showshowtheoutputvoltagedropsduringthefault,andgoes
toitsorretvaluewhenthesystemisdriventotheorretequilibriumpoint.
Ifestimationwasnotorret,therewouldbeasteadystateerror.
Finally, onean see in Fig. 1.b3) that theontrol signalis verysmooth
andiskeptinside thepresribedbounds.
Notethatduring alltime,errorsareverysmall.Theyanbemadeeven
5 Conlusions
Inapreviousworkwehaveomputedthezerodynamisofthesystemwith
respetto theterminalvoltagehavingthenobtainedahighlynonlinearse-
ondorderdynamis.Basedontypialvalues,weshowherethatthereisone
stableandoneunstablepoints,andthen,anattrationregionforthestable
one.This is amotivation to beonernedwith allthe statevetorand not
onlywiththeoutputvoltagesine,evenkeepingitregulatedtoitsreferene
value onemay nd instability for thewhole system. It is alsoamotivation
for the nonlinearontrol asthe systemmay always be driven to an unsta-
ble pointwhere alinear ontrol,speially onedesignedusing thelinearized
systemaroundthestablepoint,willnotbeabletostabilize it.
Finally,using thesameontrollerasin previousworks,weprovetheex-
ponentialstabilityofthelosedloopsystem.Wealsoprovethattheestimate
ofthe parameteronvergesexponentiallytoits truevalue.Thesystemmay
bedrivenarbitrarilyfastto thenewequilibriumpoint.Theonlyrestrition
willbethemagnitudeoftheontrolsignalandtheaeptederrorsignal.
Our present researh inludes the problem of transmission line failure.
Wehavealsostartedtheproeduretodopratialimplementationstoverify
ourssimulations.Themulti-mahineproblemwillthenbethenextstep.
Aknowledgments
TherstauthorwouldliketoaknowledgethenanialsupportofCAPES
Foundation.
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