Seventh lnteroationnl Conference om Control, Automation,
Robotics And Vision (ICARCV'OZ), Dee 2002, Slngapore
Design of Decentralized Power System Stabilizer for
Multi-Machine Power System using Periodic Output
Feedback Technique
Ftajeev Gupta', B.Bandyopadhyay",A . M . K u h n i + and T.C. Manjunath"
'Systems atid Concrol Eugineeriug,
Iiidian Institute ol Teclinolog~Bombay Munibai ISDI.4 - 400 076
E m d : [email protected], bijuan:iree.iitb.a.in. tcina,nju:he.iitb.ac.in
Department of Electrical Eugiueering,
Indian Institute ol Technolob?. Bombay: Mxnbai ISDI.4 -400 07G
Eluail: anil&ee.iitb.ac.in
Abstract
Ponw System Scabdim (PSS) are added to escitaciou
system CO euhance the darnping of electric power
system during low frequeucy oscillations. Design of
decentralized PSS for 1 machineu with 10 buses uslug
periodic output feedbad is propmed. The nonlinear
mudel of multi-mdne ryscem is linearized and linear
state space model is obtained. An output injection
gaiu is obcaiued using LQR tedmique. A decencraliaed
periodic output feedback gain n.hich realizes this outpuc
injection gaUi is obtained using LSlI approach. Tlii
method dmn'c require state of the system for feedback.
It uses only che output for f d b a d i . Thus ic is
implenientable.
Keywords: Periodic output feedback. decentralized
control, pomr system stabilizers, uiultimxhine system
1
INTRODUCTION
In the late 1980s and early 19GO's most of the
new generating units added to the electric utility
systems were equipped with continuody acting
d t a g e reghtors.
As these units became a
larger percentage of generating capacity. it became
apparent. that the voltage reguIacor action had a
detrimental impact upon the d:nanucal stabiiitJ-( or
perhaps more properly steady-state stability) of tlie
power system. Oscillations of tlie s d magmitude
and low frequency often persists for the long periods
of time and in some case it can cause limitation on
power transfer capability. Power system stabilizers
were developed to aid in damping tliese oscillation
via modulation of the generator escihtion. The
art, and science of applyhg pow-er s-ystem stabilizers
I i a been developed over the past thirty to thirty
five yeears since the first widespread application w
the Western systems of the United States. The
development tias e\~olvedthe use of various tuning
techniques and input si@ds and learning to d a l
w i t h turbine generator shaft torsional mode of
\<brations [I].
Power System Stabilizer (PSS) are added to
exitation system to enhance the damping of electric
power system during IOU- frequency oscillations.
Several methods are used in tlie design of PSSs.
Among the classical methods wed are pliase
compensation method and the root locus method.
Recently modem control methods have been used
by several researchers to take advantage of optimal
control tedmiques. These methods utilize a state
space representation of power system model and
calculate again matrix nllicli d i e n applied as astate
feedback control d l "izeaprescribed objective
function [Z].
In recent years there have been se\.eral attempts
at. designing power system stabilizer using H,
h e d robust control tedmiques. In this approach.
the uncertainty in the chosen system is modeled
in terms of bounds on frequency response. A
H, optimal controller is then synthesized which
Luarantees robust stability of the closed loop system.
However. this will lead to dynamic output feedback.
w-Mdi may be feasible. but leads to sophisticated
feedback system [3].
In practice, not all of the states are available
for measurement. In tllis case the optimal control
la^ requires M desibm the state observer. Thus
increases the iniplementation cost and reduces the
reliability of control system. There is another
&salvantage of the observer based control system.
Even a slight variations of tlie model parameters
from tlieir nominal vdues may result into si,@ficant
depdation of the closed loop performance. Hence
it is desirable to go for an output feedback desi@.
However: the static output feedbadi problem is one
of tlie nwst investigated problems in control theor?.
The complete pole assipment and buaranteed
closed loop stability is still not obtained by using
1676
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static output feedback Another approach to pole
placement problem is to consider the potential of
tiJnevar!ing periodic output feedback. It was shown
by Clrammas:and Leondes
that a controllable and
obser\able plant was discrete time pole aysipable by
periodically time-var)ing piecewise constant output
feedback Since the feedback gains are pieceuk
constant. their method could be easily implemented
and indicated a new possibility. Such a control
law can stabilize a mucli larger class of systems
than tlie staric output feedback [;]-[SI. In periodic
output feedback technique gain matrix is generally
full [6]. Tlus results in the control input of each
machine beins a function of outputs of all maclines.
Due to the geogaptucally distributed nature of
power system and lack of communication system
(unavoidable delays). the decentralized control
sdieme ma:- be more feasible than tlre centralized
control scheme. In d e c e n t r a l i i power system
stabilizei, ttie control input for eadi maclune sliould
be function of tile outbut of tliat macline only. ~ 2 i s
can be achieved by designing a decentralized PSS
using periodic output feedback technique in w1Lidi
tlie gain matrix slrould have all off-diagonal terms
zero or very small compare to diagonal tem. In
t l h t.eclmique decentralized PSS for all machine
can be desibmed in one algorithm. W PSS can b
applied simultaneously to ttie respecti\,e madline.
So the decentralized stabilizer design problem can
b translated into a problem of diagonal output
feedbad; gain matrix d e s i p for multi macline power
system.
This paper propom the design of a decetralized
power sl-stem stabiier for multi machine system
using periodic output feedback. The brief outline of
tlie paper is as follon-s: Section n presents basics of
power systen stalilizer whereas Section III contains
the modeling of muhi-macline system. Section IV
presents a brief review on decentralized periodic
output feedback control metlrod. Section V contains
tlre simulations of multi-madline potier system
at different operating points with the proposed
controller followed by the concluding section.
[4
2
2.1
REVIEW
ON
POWER
SYSTEM STABILIZERS
BASIC CONCEPT
Bas:ic function of a power system stabilizer is to
est,end stability limits by modulating generator
excitation to provide damping to tire oscillation of
synchronous machine rotors relative to one another.
The oscillations of concern is typically occur in the
frequency range-approximately 0.2 to 3.0 Ha. and
insufficient damping of these oscillation may limit
ability to transmit powr. To provide damping. the
stabilizer must produce a component of mechanical
motor speed which in phase with reference volt*
variations< For input signal. the transfer function
of the stabilizer must compensate for the gain
and plme of excitation system. the generator and
the power system. wludi collectively determine ttie
transfer function from the stabilizer output to the
component of meclianical speed.
This can be
modulated via excitation system [I].
2.2
PROCEDURE OF STABILIZER
IMPLEMENTATION
Implementation of a power system stabilizer implies
adjustment of its frequency characteristic and gain
to produce the desired damping of tlie system
oscillations in the frequency range of 0.2 to 3.0 Hz.
The transfer function of a generic power system
s t a b h e r may be e x p r e d as
where E;, represents stabilizer gain and FLLT(s)
represents combined transfer function of torsional
filter and input sibmal transducer.
The stabilizer frequency characteristic is adjusted
by vxping the time constant T.*: T1.T2.T3 and T.1
It will b noted that the s t a b i i r transfer function
includes the effect of b t l i the input signal transducer
and filtering required to attenuate the stabiIiier
gain at turbine-generator shaft torsional freqnencies
These effects. dictated by other consideration, must
be considered in addition to the %lant" [3].
3
MODELING OF A MULTI
MACHINE SYSTEM
Analysis of practical power system invotves the
simultaneous solution of equations consisting
of synchronous machines and tlie associated
excitation system and prime mo\'ers. interconnecting
transmission network. static and dynamic load
(motor) loads. and ottie devices suclr as HCDC
converters. static var compensators. "tie dynamics
of the macline rotor circuits. excitation systems:
prime mo\rer and other devices are represented
by differential equations. The r e d & is tliat the
complete system model consists of large number of
ordinary differential and algebraic equations [9].
Clavsical niodel 1.0 is aysumed for sUmdironous
macliines by neglecting damper ind dings
In
addition. the following assumptions are made for
simplicity [IO].
1.
The lo& are represented by constant
impedances.
2. p n s i e n t s saliency is ignored by considering
xq= sa
3. hfechanical power is w n m e d to be constant.
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Q'
Figpre I: Block diagram of 4 hlacliine IO bus
3.1
GENERATOR EQUATION
The machine equations ( for k'"madine ) are
p-5, = d B ( S , , - Sm*0).
I
pSme = -[-Di(Sm,
- S&)
Pm,
+
2H
3.2
- Pc,.
STATE SPACE MODEL OF A
MULTI MACHINE SYSTEM
Tire state space model of ktli macline can be
represented ay follows: [6]:[10]
ir.=[&I
q
+ [ B k ] AI^&, + AI;,).
U,
= [C,]q.
The state space model of a CmaclLine slonn in
Fig. I can be obtained using machine data. Line data
and load flow as given in [IO] as
X=
[A]x + p](AV,,,
+ AV,).
y = [CIX.
where diagonal elements (sub matrices of LX4) of
A matrix are'.il. A2. .is and .%I and off-diagonal
elements a, to all (sub matrices of LX4) depends on
the w-ay of interconnections of tlie machines.
4
4.1
ON PERIODIC OUTPUT
FEEDBACK
FLEVIEW
ON
PERIODIC
OUTPUT FEEDBACK
-
U(kT)
-
u(kr+A)
u(kr) =Ky(kr)=
:
u(kr+r-A)
The problem of pole mignment by piecenise
LOllStdnt output feedbacl. tias studled by C
"l
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z ( k ~ +
r ) = *vz(kr) +ru.
g(k)=Cr(C).
where
r = [@-IT.
....r]
Applying periodic out.put feedback in Eqn.(J). i.e..
Kg(kr) is substituted for u ( k ~ )the
. closed loop
system becomes
delays), the decentralized control scheme may be
more feavible than the Fentralized control sdieme.
Decentralized periodic output feedback control can
be achieved by making the off diagonal elements of
ICo. I i l . ...... K3y-1 matrices zero.
Non: tlie problem can be formulated in the
framework of Linear hiatrh Inequalities using
Eqns.(S)-(9) and the desired gain matrices can h
obtained.
5
z(kr+r) =(9"+rKC)r(tr).
Tlie problem liav now taken tlie form of static
output, feedback problem. Eqn.(B) suggest that an
output, injection matrix G be found such that
p((@
+ GC) < 1.
(6)
diere p ( ) denotes. the spectral radius.
By
obsermbility one can clioose an output injection,
gain G to achie%reany desired self-conjugate set of
eigenvalues for the closed loop matri. (0" GC) .
and from N Z I I it follows that one can find a
periodic output feedback gain wlucli realizes the
output injection gain G by solving
+
r K = G.
(5)
for K.
The controller obtained from the a h v e equation
nil1 give desired behavior: but might require
excessive control action. To reduce tliy effect we
relax the condition that K esactly satisfy the above
linear equation and include a constraint on the gain
K. Tllus we arrive a t the following in equations
The 4 madlines. 10 bus hiulti- hiadline Power
System data are considered for desibping periodic
output feedlmck controller uying Lhff approach of
hiatlab softwaw. The single l i e diagram of the
system is shown in Fig.1. The maelline data. line
data. A\= data and load flow are given in [IO].
The a h v e multi-machine system is modelled
us& Simulink Toolbox of Matlab and a Linear state
space model is obtained for the same. Then discrete
model is obtained for tlie sampling time r = 0.01 sec.
Using the method discussed in Section 4
stabilizing output injection gain matrix G (16x4)
is obtained.
Using LkD approach. Eqns.(&S)
are solved using different valuer of p to fmd tie
decentralized gain matrix K.
The decentralized periodic output feedback gain
matrix I< ( 16x4)is obtained av @\,en in appndkx B.
The closed loop responses under tlie control lau:
Eqn.(J) nitli tlus decentralized gain K for linearized
model of all four machines are satisfactory and able
to stabilize the outputs.
+rKC) are found to h
The eigen values of (afiwithin the unit circle as given in appendix B.
11ICI1 < PI
6
[ $*
-p;I
[ (rK - G)T
y
<
]
<0
(9)
In tiis form the L.hU Tool Bm Matlab can be used
for syntliesis [II].
4.2
SIMULATION WITH NON
LINEAR MODEL
A simulink based b~odi.'diapamincluding a~ tile
nonlinear block is generated. The slip of the
madline is talien ay output. Tlie output slip signal
with decentralized gain 1%-and a limiter is added
to V,,! signal which is used to provide additional
damping. Tllis is used to damp out the small
sibmal disturbances via modulating the generator
excitation. The output must be limited to prevent
the PSS acting to counter action of A\X.
Simulation results of different b-nerators are
shown in Fig2 without controller and with controller
and the same with fault are shown in Fig.3 without
controller and with controller.
0
y - G )
NUMERICAL EXAMPLE
(6)
DECENTRALIZED PERIODIC
OUTPUT FEEDBACK
In periodic output feedback for multi macline
system. gain matrix is generally full [GI. Tlis
results in the control input of each machine being
a function of outputs of all machine% Due to the
ge~gaphicallydistributed nature of power system
and I d of communication system (unavoidable
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2'
-2'
=*"
GENERATOR <
ro4
2
90-
.
n-
6
in --nd.
a
l
10
GENERATOR 2
GENERATOR 4
.
6
rim. h wlond.
8
,o
Fibwe 3: Open and closed loop responses with
fault using decentralised periodic output feedback
controller
Fig.m 2: Open and c l o d b o p responsev using
decentralised periodic output feedbadi controller
1680
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7
.., .. ..
..
. ,
..
.._.
... ..~
_i.
. .,
...
. ~.. ~..,
. .
.
.
CONCLUSION
The linearized model of multi-madune( 4 generators)
power system %it11 10 buses is constdered here. Tlie
slip si@ is talien a4 output. Tlie decentralized
periodic output feedLa& control is desiped and
applied to tlie linear as well as nonlinear system. The
siruulatiou here sl~o~vs
that the decentralized periodic
output fedback control technique can be used to
design power system stabilizer for niulti-macline(
4 generators) paver s.%tem with 10 buses. The
control input for these generators are required of
small ma64tudes. The method is more general than
static output feedbark and also leads to decent,raliwd
control for multi-miline system . The decentralized
control can Le applied simultanmusly to tlie all
maclines. The input applied to each macllie is a
function of output of the respective madline. Thus
the applied control scheme is decentralized in nature.
References
[I] E.V.Larsen and D..A.Swann: a.%pplying Power
System Stabilizers Part- I: General Concepts”.
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301i-3024.
0
[Z] E.V.Larsen and D . . i . S w m : u.%ppl?.ing Power
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+
+
0.48Mi30.4501 - 0.4SMi. 0.5117 0.43S6i,
0.511i 0.43S6i 0.8201 0.4326,0.5201 - 0.4324i,
0.9092 0.5006il 0.9092 - 0.5006i. 0.6215 0.15621,
0.6215 - 0.1562i. 0.6333 0.1239i. 0.6333 - 0.123%
i.nonn, 0.01-i; O.O%;i, n.siii - 0 . o s i i . O.91iI) ]
[0.4501
-
+
+
+
[GI Rajeev’ Gupta. B. Bandyopdlyay and
h.hI.Kukami: -Design of Power System
Stabilizer for Multi-Madline Power System
using Periodic Output Feedback Tedmique”.
International conference on Quality, Reliability
and Conhol, ICQRC2001, Mumboi, Indio. pp.
C46-I-8.
+
4684
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+