A New Stabilizer and Design Algorithm to Minimize the Excitation of Undesirable Oscillations

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1 Tuesday, August 14, 2007

A NEW STABILIZER AND

DESIGN ALGORITHM TO MINIMIZE THE EXCITATION OF

UNDESIRABLE OSCILLATIONS

N. Kshatriya, U.D. Annakkage, A.M. Gole


OUTLINE

Ø Background

Ø Partial Left Eigenstructure Assignment Technique

Ø SS type PSS

Ø Design Algorithm

Ø Results

Ø Conclusions

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POWER SYSTEM STABILIZERS

Traditional PSS Design Objective

Exciter Ge(s)

Generator Gg(s)

1 2Hs

1 s

∆ωr

∆Te

∆Tm

+ _ ∆δ Vref

PSS K*Gp(s)

Power System

Et,

θ

Pg, Qg Et

Vpss

Proportional to ∆ωr

Exciter Ge(s)

Generator Gg(s)

1 2Hs 1

2Hs 1 s 1 s

∆ωr

∆Te

∆Tm

+ _ ∆δ Vref

PSS K*Gp(s)

Power System

Et,

θ

Pg, Qg Et

Vpss

Proportional to ∆ωr

Re(λ)

Im(λ)

Damp=0.05

x x

Improve damping of poorly damped mode

Re(λ)

Im(λ)

Damp=0.05

x x

Make unstable mode sufficiently damped


WHY EIGENSTRUCTURE?

( ) ( ) ( ) x t A x t B u t •

= +

Linear System Dynamics

( ) (0) t x t V e W x Λ =

Time domain solution

V = right eigenvector matrix

W = left eigenvector matrix

Λ = diagonal matrix of eigenvalues

x(0) = initial condition

Ø Traditional PSS and design algorithm is concerned about closedloop eigenvalues only

Ø Eigenvalues decides rate of decay

Ø Right and Left eigenvectors together decide the shape of the response

Ø Further improvement in dynamical performance can be achieved if eigenstructure is considered in design algorithm

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RIGHT AND LEFT EIGENVECTORS

LEFT EIGENVECTOR

Ø Decides excitation of mode : y i (0) = w i ix(0)

Ø Judicious selection can minimize excitation of mode

RIGHT EIGENVECTOR

Ø Decides presence of mode in states: x(t) = Σ v i y i (t)

Ø Judicious selection can minimize presence of mode in states

Proposed New algorithm optimally selects left eigenvector

( ) (0) t x t V e W x Λ =

Time domain solution


EIGENSTRUCTURE ASSIGNMENT TECHNIQUE

Ø Suitable for MultiInput, MultiOutput (MIMO) LTI system

Ø First theoretical application in 1970’s

Ø Many techniques for controller design based on ESA

Ø Application : Aircrafts, helicopters, missiles, etc.

Ø Not yet found its place in power system

Ø Proposed algorithm is based on Partial LESA technique

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CONTROL SYSTEM

LTI System

LINEAR output feedback control system

( ) ( ) ( ) ( ) ( ) x t A x t B u t y t C x t

= + =

g u(t)

Linear Controller

K

y(t)

states: ; inputs: ; outputs:

n m r

states: a

states: ; inputs: ; outputs:

n n a m m a r r a

= + = + = +

G F K

E D

=

LTI System

( ) ( ) ( ) ( ) ( ) x t A x t B u t y t C x t

= +

=

g

( ) ( ) ( ) ( ) ( ) ( )

z t D z t E y t u t F z t G y t

= + = +

g

DYNAMIC output feedback control system

Dynamic Controller

( ) u t ( ) y t


PARTIAL LESA : PARAMETRIC SOLUTION

( ) ( ) ( ) ( ) ( ) ( ) ( )

x t A x t Bu t y t C x t u t K y t

= + = =

g

states: ; inputs: ; outputs: n m r

LINEAR output feedback control system

( ) ( ); C

C

x t A x t A A B K C

=

+

g

@

ClosedLoop System

Problem Statement

If meigenvalues λ 1 ,…,λ m are to be assigned to Ac then find parametric solution of associated left eigenvector and K.

Solution* Parametric Vectors

1 , , m g g K 1 ( ) T T i i i n w g C I A λ − = −

1 ( ) m m K W B G − =

Left eigenvector Controller

1 1

;

T T

m m T T

m m

w g W G

w g

= =

M M

*M.M. Fahmy, J. O'Reilly, “Multistage Parametric Eigenstructure Assignment by OutputFeedback Control,” Int. J. Control, vol. 48, no. 1, pp. 97116, 1988

r i i g if λ ∈ ∈ ¡ ¡

, * , * r i i i i g g if λ λ ∈ ∈ £ £

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STATESPACE TYPE PSS

LTI System

( ) ( ) ( ) ( ) ( ) x t A x t B u t y t C x t

= +

=

g


( ) ( ) ( ) ( ) ( ) ( )

z t D z t E y t u t F z t G y t

= +

= +

g

Dynamic Controller

states: a

( ) ( ) ( ) ( ) ( ) x t A x t B u t y t C x t

= + =

g LTI System Linear Controller

( ) ( ) u t K y t G F

K E D

=

=


Ø Total degrees of freedom available are m*r

• Independently assignable elements of (m x r) matrix K

Ø Degrees of freedom utilized as:

• rdegrees of freedom are utilized in assigning eigenvalues

• r*(m1) degrees of freedom are utilized in assigning associated left eigenvector

Ø A leadlag type PSS can be formulated as dynamic controller

• Independently assignable elements will be less than m*r

• Degrees of freedom in assigning eigenvalues and/or left eigenvector must be sacrificed


ALGORITHM BASED ON PARTIAL LESA

1. Linearized system around an operating point

• states: ; inputs: ; outputs: n m r

2. Selectm eigenvalues, Λ o,m ,those are to be assigned new values.

• Poorly damped modes or unstable modes

• Most sensitive to controller

3. Calculate dimension of dynamic compensator states

• a m m = −

4. Selectm eigenvalues, Λ m , to be assigned to Λ o,m

5. Optimize parametric vectors g 1 , … ,g m using nonlinear optimization

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NONLINEAR OPTIMIZATION

1 1

m Ti i

i

w t φ =

= ∑ g Ø Minimize weighted sum of left eigenvector :

Objectives

Constraint

max 2: max ( ) C abs K < k Ø Controller parameters should be reasonably small :

min

min

min ( ) 1:

minRe( ) i

i

damp C

λλ

>< dr Ø Unassigned closedloop eigenvalues are acceptable :

2 , 1

( ) m

T i i i o i i

i w BKCv φ λ λ

=

= − + ∑b Ø Λ o,m Λ m :


TEST SYSTEM

ØEach generator represented using 2axis model ØEach generator equipped with an exciter (type AC4A)

Bus1 Bus2

Bus3 Bus4

Bus5

Bus6

Gen.1 Gen.2

Gen.3 Gen.4

Area1

Area2

0.0025+j0.025

0.0025+j0.025

0.001+j0.010

0.001+j0.010

0.022+j0.22

1.02 pu 1.01 pu

1.01 pu 1.02 pu

1400MW 250MVAR

254MVAR

900MW 250MVAR 255MVAR

400 MW

EigenAnalysis (No PSS)

ω4

ω2

δ3

Dominant State

Area2 Plant

Area1 Plant

InterArea

Mode type

19.08 1.1992 1.4643±j7.5345 3

16.13 1.2303 1.2633±j7.7302 2

2.80 0.3352 0.059±j2.1062 1

Damping Ratio (%) Freq. (Hz) Eigenvalue No.

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CASE 1

1 K s T s +

1

3

1 1

T s T s

+ +

2

4

1 1

T s T s

+ +

u y

Washout Filter

LeadLag Blocks

Input

1. LeadLag type PSS

Ø Designed using traditional phasecompensation technique

3. Input

Ø Speed Ø Generator power

2. StateSpace type PSS

Ø Designed using proposed algorithm

( ) ( ) ( ) ( ) ( ) ( ) z t D z t E y t u t F z t G y t

= + = +

& Input

Washout Filter

StateSpace PSS y

u 1 T s T s +


EigenvalueAnalysis

RESULT : SPEED INPUT…

1.6±j7.5345 1.4677±j7.5552 1.4643±j7.5345 3 1.2624±j7.7335 1.2632±j7.73302 1.2633±j7.7302 2 0.1058±j2.1073 0.1058±j2.1063 0.059±j2.1062 1

SS PSS LeadLag PSS NoPSS No.

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RESULT : SPEED INPUT… Nonlinear simulation : 20 ms threephase fault at Bus1


RESULT : POWER INPUT…

EigenvalueAnalysis

1.6±j7.5345 1.6023±j7.9486 1.4643±j7.5345 3

1.2635±j7.7306 1.2638±j7.7315 1.2633±j7.7302 2 0.1058±j2.1073 0.1052±j2.1042 0.059±j2.1062 1

SS PSS LeadLag PSS NoPSS No.

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RESULT : POWER INPUT Nonlinear simulation : 20 ms threephase fault at Bus1


CASE 2 1. Dual input LeadLag type PSS : PSS2A

Ø Designed using traditional phasecompensation technique

2. Dual Input StateSpace type PSS

ω Gen4

Pe Gen4

M=5 N=1

Filter and Transducer

Assumed to be known

ω Gen4

Pe Gen4

M=5 N=1

ω Gen4

Pe Gen4

M=5 N=1


Assumed to be known


Assumed to be known

To be designed Assumed to be known

Filters and Transducers

3 z∈ ¡ sz Dz Ey u Fz Gy

= + = + u

1

1 1 w

w

sT sT +

2

2 1 w

w

sT sT + 6

1 1 sT +

3

3 1 w

w

sT sT +

4

4 1 w

w

sT sT +

2

7 1 s K sT +

y

ω Gen4

Pe Gen4

D(3x3), E(3x2), F(3x1), G(2x1)

To be designed To be designed Assumed to be known


Assumed to be known



= + = + u

1

1 1 w

w

sT sT +

2

2 1 w

w

sT sT + 6

1 1 sT +

3

3 1 w

w

sT sT +

4

4 1 w

w

sT sT +

2

7 1 s K sT +

y

ω Gen4

Pe Gen4

D(3x3), E(3x2), F(3x1), G(2x1)


= + = + u

1

1 1 w

w

sT sT +

2

2 1 w

w

sT sT + 6

1 1 sT +

3

3 1 w

w

sT sT +

4

4 1 w

w

sT sT +

2

7 1 s K sT +

y

ω Gen4

Pe Gen4

D(3x3), E(3x2), F(3x1), G(2x1)

sz Dz Ey u Fz Gy

= + = + u

1

1 1 w

w

sT sT +

2

2 1 w

w

sT sT + 6

1 1 sT +

3

3 1 w

w

sT sT +

4

4 1 w

w

sT sT +

2

7 1 s K sT +

y sz Dz Ey u Fz Gy

= + = + u

1

1 1 w

w

sT sT +

2

2 1 w

w

sT sT + 6

1 1 sT +

3

3 1 w

w

sT sT +

4

4 1 w

w

sT sT +

2

7 1 s K sT +

y

ω Gen4

Pe Gen4

D(3x3), E(3x2), F(3x1), G(2x1)

Ø Designed using proposed algorithm

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CASE : 2

Real Imag. Real Imag. 1 0.1058 2.1073 0.1068 2.1063 2 1.2633 7.7306 1.2634 7.7311 3 1.5552 7.5940 0.3267 1.5863 4 0.2493 0.3802 1.6000 7.7500 5 0.6301 0.9345 0.2571 0.3976 6 0.2872 0.3916 0.1579 0.2435

Mode PSS2A SS PSS

EigenvalueAnalysis


Comparison of speed transient for 20 ms 3Phase fault at Bus1

CASE : 2

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CONCLUSIONS

Ø Selecting optimum weights in optimization is critical

Ø Objective : Minimize excitation of lowfrequency lowdamped mode

Ø Proposed

• StateSpace Type PSS

• Design algorithm based on Partial Left Eigenstructure Assignment technique

Ø Results depends on type of input and number of inputs

• Speed input : No improvement

• Elect. Power : Excitation of interarea mode is minimized to a good extent

• Dual Input : Interarea mode is completely suppressed


THANK YOU!!

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Acceptable Unacceptable

Re(λ)

Im(λ)

x

*

* x

x x

x

*

* x

x x

PRECISE EV ASSIGNMENT

*, * :OPENLOOP eigenvalues x : Assigned eigenvalues


Not acceptable

CLOSEDLOOP EV CONSTRAINT

x

x

*

*

*

*

*

*

* x

* *

x

* *

*

Acceptable Unacceptable

Re(λ)

Im(λ)

*, x :OPENLOOP eigenvalues

x : Concern lowdamped mode

*, x :CLOSEDLOOP eigenvalues

x : Concern lowdamped mode

Documents

A New Stabilizer and Design Algorithm to Minimize the Excitation of Undesirable Oscillations