Stabilization of Multimachine Power Systems by Decentralized

Feedback Control

Zhi-Cheng Huang

Department of Communications, Navigation and Control Engineering

National Taiwan Ocean University

1

Outline

Introduction

Decentralized controller design

Illustrative example

Conclusions

2

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

3

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

4

Decentralized controller design

Salient-pole synchronous generator

''

1( ) [ ( ) ( )] qi i fi qi

doi

E t E t E tT

Generator electrical dynamics

5

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

6

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

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

8

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

9

Decentralized controller design

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

10

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

11

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

An Illustrative Example

A three-machine example system is chosen to demonstrate the effectiveness of the proposed nonlinear decentralized controller

13

An Illustrative Example• The excitation control input limitations

• The generator #3 is an infinite bus and use the generator as the reference

14

3 ( ) ( ) 6 fi ci fiE t k u t,

1,2i

'3 const. 1 0 qE

System parameters

15

An Illustrative Example

An Illustrative Example

A three-machine power system

16

An Illustrative Example• The DFL compensated model for the

generators #1 and #2

17

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

An Illustrative Example

• with

18

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

An Illustrative Example• Assume

19

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

An Illustrative Example

• the DFL compensated power system model will be globally asymptotically stable by the linear local state feedback

20

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

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

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

22

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

23

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

24

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

25

Q&A

The End

Thanks for your Attention

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