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Stabilization of Multimachine Power Systems by Decentralized
Feedback Control
Zhi-Cheng Huang
Department of Communications, Navigation and Control Engineering
National Taiwan Ocean University
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Outline
Introduction
Decentralized controller design
Illustrative example
Conclusions
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Introduction
State-dependent impulse disturbance will be investigated
Direct feedback linearization compensator will be proposed
Boundedness of the system states will be guaranteed within the derived impulse intervals
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Decentralized controller design
n synchronous machines
( ) ( ) i it t
0( ) ( ) ( ( ))2 2
ii i mio ei
i i
Dt t P P t
H H
Mechanical equations
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Decentralized controller design
Salient-pole synchronous generator
''
1( ) [ ( ) ( )] qi i fi qi
doi
E t E t E tT
Generator electrical dynamics
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Decentralized controller design Electrical equations
( ) ( )fi ci fiE t k u t
' '
1
( ) ( ) ( )sin( ( ))
n
ei qi qj ijj
P t E t E t t
' '
1
( ) ( ) ( ) cos( ( ))
n
ei qi qj ijj
Q t E t E t t
'
1
( ) ( ) cos( ( ))
n
di qj ijj
I t E t t
'
1
( ) ( ) sin( ( ))
n
qi qj ij ijj
I t E t B t
( ) ( )qi adi fiE t x I t
' '( ) ( ) ( ) ( ) qi qi di di diE t E t x x I t
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The compensated multimachine power system model
0( ) ( ) ( ( ))2 2
ii i mio ei
i i
Dt t P P t
H H
' '
1 1( ) ( ) ( )
ei ei fidoi doi
P t P t v tT T
' '
1
( ) ( ) sin( ( ))
n
qi qj ij ijj
E t E t B t
' '
1
( ) ( ) cos( ( )) ( )
n
qi qj ij ij jj
E t E t B t t
( ) ( ) i it t
where ( ) ( ) ei ei mioP t P t P
'( ) ( ) ( ) ( ) ( ) ( ) fi qi ci fi di di qi div t I t k u t x d I t I t' ( ) ( ) mio doi ei iP T Q t t 7
Decentralized controller design
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Decentralized controller design
DFL compensating law
'1( ) { ( ) ( ) ( ) ( )
( ) fi fi mio di di qi di
ci qi
u t v t P x x I t I tk I t
' ( ) ( )} doi ei iT Q t t
( ) 0qiI texcept for the point (which is not in the normal working region
for a generator)
where
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Generalized uncertain DFL compensated model ( ) [ ( )] ( ) [ ( )] ( ) i i i i i i fix t A A t x t B B t v t
1 2( , ), ( , ) ij i j ij i jx x x x
1 1 1 2 2 21 1{ ( ) ( , )} { ( ) ( , )}
N N
ij ij ij i j ij ij ij i jj jG t g x x G t g x x
where
1 2( , ), ( , )ij i j ij i jg x x g x x
known real constant matrices and controllable real time-varying parameter uncertainties
interaction terms
unknown nonlinearity
constant with values either 1 or 0
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Decentralized controller design
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Decentralized controller design
Assumption 1. (System Matrix Uncertainties)
1 2( ) ( ) ( ) i i i i i iA t B t L F t E E
with Lebesgue measurable element
( ) ( ) Ti iF t F t I
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Decentralized controller design
Assumption 2. (Interaction functions)
1 1 1 1 1ˆ( ) ( , ) ( ) ( , )ij ij i j ij ij ij i jG t g x x M C t g x x
with Lebesgue measurable element
1 1 2 2( ) ( ) , ( ) ( ) T Tij ij ij ijC t C t I C t C t I
2 2 2 2 2ˆ( ) ( , ) ( , )ij ij i j ij ij ij i jG t g x x M C g x x
1 1 1ˆ ( , ) ( ) ( ) ij i j ij i ij jg x x R x t W x t
2 2 2ˆ ( , ) ( ) ( ) ij i j ij i ij jg x x R x t W x t
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Assumption 3. (impulse disturbance)
( ) ( ) ( , ( )), i k i k i k i k kx t x t w t x t t
where
( ) ( ) ( ) ( ) T
i i i eix t t t P t
( , ( )) ( )i k i k ik i kw t x t x t
0( ( , ( )) ( ) ( ) ) i k i k ik i k i ik i k ikw t x t x t x x t
( , ( ))i k i kw t x t the effect of state changing
0 0 , Tio mio ik ik miox P P
with12
Decentralized controller design
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An Illustrative Example
A three-machine example system is chosen to demonstrate the effectiveness of the proposed nonlinear decentralized controller
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An Illustrative Example• The excitation control input limitations
• The generator #3 is an infinite bus and use the generator as the reference
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3 ( ) ( ) 6 fi ci fiE t k u t,
1,2i
'3 const. 1 0 qE
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System parameters
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An Illustrative Example
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An Illustrative Example
A three-machine power system
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An Illustrative Example• The DFL compensated model for the
generators #1 and #2
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1 1 1 1 1 1 1( ) ( ( )) ( ) ( ( )) ( ) fx t A A t x t B B t v t
112 1 2 211 1 212 2sin( ( ) ( )) ( ) ( ) G t t G t G t
2 2 2 2 2 2 2( ) ( ( )) ( ) ( ( )) ( ) fx t A A t x t B B t v t
121 2 1 221 1 222 2sin( ( ) ( )) ( ) ( ) G t t G t G t
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An Illustrative Example
• with
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1
0 1 0
0 0.625 39.27
0 0 0.1449
A
,
1
1
0 0 0
( ) 0 0 0
0 0 ( )
A t
t
1
0
0
0.1449
B
, 1
1
0
( ) 0
( )
B t
t
2
0 1 0
0 0.2941 30.8
0 0 0.1256
A,
2
2
0 0 0
( ) 0 0 0
0 0 ( )
A t
t
2
0
0
0.1256
B
, 1
2
0
( ) 0
( )
B t
t
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An Illustrative Example• Assume
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the operating point is 0(0,0, )miP with
10 1.1 . .mP p u , 20 1.0 . .mP p u
impulse disturbance is (2) with
0.5(1 ) kk e for 100k
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An Illustrative Example
• the DFL compensated power system model will be globally asymptotically stable by the linear local state feedback
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1 1 1 1 1 1 101
1( ) { ( ) 1.606 ( ) ( ) 6.9 ( ) ( ) }
( ) f f q d e m
q
u t v t I t I t Q t t PI t
2 2 2 2 2 2 202
1( ) { ( ) 2.041 ( ) ( ) 7.96 ( ) ( ) }
( ) f f q d e m
q
u t v t I t I t Q t t PI t
with
1 1 1 1 10( ) 46.7046 ( ) 71.6728 ( ) 241.2868( ( ) ) f e mv t t t P t P
2 2 2 2 20( ) 57.0220 ( ) 90.8554 ( ) 315.7032( ( ) ) f e mv t t t P t P
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Assumptions
the impulse instant kt is chosen randomly
between 0sec and 0.15sec the impulse disturbance is in the form of
( , ( )) 0.05( 1) (1 )sin( ( ) ( ) k kk k k kt t e t t
initial points 1(0) 0.5rad , 1(0) 0.2rad/s , 1(0) 0.5 . .eP p u
,
'1(0) 1.03 . .qE p u
,
2 (0) 1rad , 2 (0) 0.5rad/s , 2 (0) 0.5 . .eP p u,
'2 (0) 1.01 . .qE p u 21
An Illustrative Example
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An Illustrative Example
0 5 10 150
5
time (sec)
δ i(t)
(rad
)
0 5 10 15
-4
-2
0
time (sec)
ωi(t
) (r
ad/s
)
0 5 10 15-0.2
0
0.2
0.4
0.6
time (sec)
Pei
(t)
(p.u
.)
Fig. 1. The state responses with finite number of impulse disturbances
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An Illustrative Example
Consider the equidistant impulse
disturbance with 0.05 ik The impulse disturbance
( , ( )) 0.05( 1) sin( ( ) )( ( ) ) ki k i k i k io i k iow t x t x t x x t x
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An Illustrative Example
0 5 10 15 20 25 30
0
0.5
1
1.5
time (sec)
δ i(t)
(rad
)
0 5 10 15 20 25 30-1
0
1
time (sec)
ωi(t
) (r
ad/s
)
0 5 10 150.5
1
1.5
time (sec)
Pei
(t)
(p.u
.)
Fig. 2. The state responses with impulse disturbances
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Conclusions
The problem of decentralized control of multimachine power systems with state-jump disturbances has been explored
A new synthesis algorithm for the direct feedback linearization compensator has been proposed
Sufficient conditions have been derived such that the decentralized practical stability can be guaranteed
The states of the uncertain multimachine power systems with equidistant or periodic impulse disturbance will attract into a bounded ball
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Q&A
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The End
Thanks for your Attention
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