Parameter Adjusting of a Static Compensator Based on Dynamic Equivalent Model of a Large Power System

Copyrigh t © IFAC Electric Energy Systems Rio de Janeiro, Brazil. 1985

STATIC VA R COMPENSATION

PARAMETER ADJUSTING OF A STATIC COMPENSATOR BASED ON DYNAMIC EQUIV ALENT MODEL OF A LARGE POWER SYSTEM A. M. da Silveira* W. S. Mota** and J. C. de Castro** * Fullda ((io L'1I;"I'nidadl' F l'dl'l"a/ do ,\lamllitrio ** L'II;"I'nidadl' F edI'm/ do Pami!Ja

Abs tract . This ~Qrk presents a technique for parameters adjusting of a static compensator based on a linearized model in state space form of a compensator connected to a dynamic equivalent model . The dynamic equivalent model is used to reduce the power system model order computed by identification theory. Using the interconnected model ie static compensator connected to the dynamic equivalent model, optimization techniques are applied for the compensator gains and time constant adjus ting. For illustration purposes of the techniques here proposed , an application is performed to a realistic p~ver system , This system consists of the west System of CHESF (Brasilien Power Company) Keywords. Control system analysis; dynamic response; parameter estimation; optimal search techniques.

IN1RODUCTION Reactives compensation using high poIVer thyristors technology has been recently used by Industries and Power Systems for voltage stabilization, power fac tor correction and flicker reduction. The static compensator is the responsible element for this type of compensation in Power Systems, These equipments require gains and time constants adjusting for a satisfactory dynamic behavior of the system under perturbations, By trial and error method the analysis and parameter tunning need a lot of computer simulations of the IVhole system in a transient stability program; specially when the power system is large, Trial and error method using classic control techniques is too limited because it can only be applied to selection of a reduced number of parameters , Analysis of systems in state space form also needs a strong computational effort, In this I;ork, the size of the poI;er system is reduced by using a dynamic equivalent model to repre sent the system (Nota, 1981). TIlen the static com-=pensator lineari:ed model is connected to the equi\' alent model to form an interconnected model of 10li order in state space form. On this model the adjusting of selected parameters are performed. The adjusting technique is based on a suggestion made by, (Castro, 198~) , consisting of an optimi:ation process of system controllers I;here the selection of adjustable parameters are made simultaneously . DYWIIC EQUI\'ALEl\! mDEL Here, a d)~amic equivalent model in state space form, identified by the recursiYe least squares method (~Iota, 1981) is used to represent part of a large pOI;er system, TIle dynamic equi\'alent model h'ill ha\'e the fol101;-

ing final form: r:,X

A r:,X e e Cer:, Xe

e

r:, I

+

B

r:, y

(1)

+

D r:,V e

(2)

E'

where

v;

input vector

[r:,vDl r:,v Q

r:,I;

[::~1

output vector

(See figure 1) I,ith r:,v

D

f:,\'Q r:, i D r:,iQ

vDo

VD \'Q iD

VQo i Do

iQ

iQo

TIle subscript ting point.

"0"

corresponds to the initial opera -

For application purposes, consider the realistic poI;er system of figure 2 consisting of the h'est system of "CHESF" (Brasilien POKer Company), Khere a static compensator is located at bus - , The remaining system (i,e , I;ith bus - excluded) is represented by a d~~amic equi\'alent model evaluated by identification techniques. To eliminate the d)~am­ ic effects of the static compensator in the identification process, it I;as represented by a large inertia. small impedance generator, in a transient stability program. Initially four single input. single output discrete transfer functions relating the real and imaginary components of current and \'oltage at the compensa tor terminals . are identified. and

A. M. da Silveira. W. S. Mota and J . C. de Castro

218

A third order numerator and denominator were found to be satisfactory in the identification of each transfer function. However. more accurate results can be obtained by higher order transfer functions . ~ b + b Z- 1 + b Z-2 + b Z- 3 3 o l 2 -1 Z2 + a Z- 3 u (Z) 1 + a l Z + a2 3 To facilitate the parameter identification precess. the algebraic component bo was not identified. This component was computed l ater. and incorporat ed into the model in state space form. This value can be calculated as the ratio between the output and input of the initial values of the dynamic simulation . lIiD For the transfer function • the parameter vector lIVD T a = [ -al' - a 2 • - a 3 ' b l • b2 • b 3 ]

program. Linearized model A linearized model in state space form can be ob-

tained from the block diagram Fig. 7.

The linearized state equations around an operating point in per unit values are: G1RC (3) lIw lIXCl TMRC 1 l4 ) lIX C2 = - - (LlXCl - lIXc2 + G1RC. lIw) TFJ{C where

a

=

[2.94 73 - 2. 8989 0.9516 - 0.0671 0. 131 3

-o.o64d

6V - GKRC .

lIw

1II

+

lIu

Wlth vDo

lIV

was identified by forcing an deterministic small variation in the real part of the infinite bus I t erminal voltage and computing the real part of the corresponding terminal current during the simula tion in a transient stability program. The following numerica l values were obtained :

as follows:

=---

I Vo I

1I1

lIvD

+

MD

+

~ IVo l

iDo

=---

LlV Q

iQo II 0 I

110 1

MQ

The net current through the compensator is given by

M

Similary. for the transfer function ~. the fol lowing parameter vecto r was comput ed lIvu a

= [ 2. 875 1

For the trans fer function ~ • the parameter vec tor was identl fied by lIvO forcing a small variation in the imaginary component of the voltage The fo llowing numerical val ues were computed : a

=

or in complex form

- 2. 7667 0.8907 - 0. 0930 0 .1 837 _0. 0917]T or (5 )

[2. 9723 - 2. 952 1 0. 9797 -0 . 10495 0. 1995 _0. 0948]T

Similary. for the transfer function lIiQ • the fo l10\ving parameter vector was lIvQ computed:

(6 )

In order to interconnect the compensator with the system. the resultin~ comoensat or admittance should be expressed in per unit of the transmi ssion system bases . trans~i ssjon

a

=

[2.9562 - 2. 9198 0. 9635 0.1 0903 - 0. 2095 0.1 007]T

The figures 3. 4 .5 and 6 show the response of the original system and ident ified model respectively for each transfer function. :-Jm,' . the transfer functions are represented in s t ate space form . and . using the superposition theorem the four single input. single output models are connect ed to form the discrete dyl1amic equivalent syst em. Finally . the discrete model is trans formed into t he corresponding continuous linear model. (Brogan . 1974) . in t he form

1I1

'''V''\

+

Be

1I\'

Ce "'\

+

De

lIr

\,'ere lIX = (LlXl . 6X2 .... .!lX12 )T . s t ates of the equivalefit model. sn.TIC

CD~!PE\S..l.roR ~ IODELS

For convention. inductive reactante has negative value . Then. from the block diagram one can have the fo llowing equation : (7)

Subs t i tuting (7) into (5) and (6) (8) (9)

A linearized \'ersion of equat ions (8) and (9) is:

lIid .!li

Q

-vQo lI\Z \'Do

6XC2

+

Y

0 lIVq Y 0 ':'\'D

(10 ) (11)

\,here

\onlinear model The static compensator model used for this \\ork \,as supplied b:' "CHESF" . The figure ~ sho\,s the block diagram used for nonlinear d}l1amiC sumulation in a transient stability

\ow . after some substi t utions . the state equations (3) and (.1) can be \,Titten in matrix notation

2 19

Parameter Adjusting of a Static Compensator

lIXC1 lIX C2

t

GMRC

+

_ _ 1_) TFRC

( CMRCIVo l TFRC Yo

TFRC

GKRC

lMRC I I o I

~C

. GKRC i

1101

+

Qo

Interconnected model of the static compensator and dynamic equivalent

After the combination of state equations,

lI~C ~

]

CMRC.vDo

(lIVS) input vector

~

The dynamic equivalent model equations (1) and (2) may be combined with t he s t at ic compensator model equation (12), (13) and (14) as follow:

. VQo lMRC . Ivo I y0

+

+

l lMRC

flXC2

J

CMRC

" IDo

lIu

lIX C1

CMRCIVol lMRC Yo

0

lIXe

]

[

lMRc.lvolyo

[ AC 0 ]. [ flXc 0 Ae lIXe

BoC ]

1

+

[ cc 0

0 ]. Be

r

1II

lIV

lIu

1

+

(15)

+

(}IRC . GKRC " [ TFRC 11 I .IQo o

+

Also for the algebraic equations

CMRC . VDo ] TFRc.lv Iy 0

[::]'[:c :'ll:~H:c :'][:: j

0

(}IRC

nlRc

lIu

Substituting (16) into (15)

GMRC TFRC

[::1_ [~~CCNDC

The equatIons (10) and (11) can be written in matrix notation for lIVas:

lIV

[

.[::: 1

[ .tin

(16J

0

vDo/yo

0

VQo/yo

1

0

C2

1

+

· [:c 1

'"

where

[ "D

- ';'0 1

l / Yo

[ "et lIX

Now, the interconnected model has the followine form:

lIiQ

equation for the compensator output may be

A

lIy

C

lIX lIX

+

B lIu

(17)

where or [ 0

1

fi X ~ (lIXC1 ,6XC2 ,6X1 ,6X2 , .. . lIX12 )T

1

6u \0\" the lineari:ed model of the static compensator has the form ~\

- C ~r

.jy

~

~

~

LXC

+

DC ':'XC

+

"\:

BC

lIu

+

Cc

1II

EC 1II

FC t Xc

and C

~

[ 0 1 0 0 0 0 0 0 0 0 0 0 1

( 12)

Adjustable parameters for the model (13) ( 1-1 )

The parameters of the model that may be adjustable are Gt-!RC, GKRC and DIRC; which "ill be denoted by 11 1, nz and 11 3 respectively ,,"here

Khere

lIVS

11

~

(11 1 ' n 2 and

'I

3)

2XC

(2 XC1 ' ~XC2 ) T state vector

adjustable parameters vector.

.!l\"

(.c;vD , tn"Q ) T voltage vector

Then, equation (17) will be represented in the form

!cl 21"

(ti D , -'. iO ) T current \"ector ( ':'XC ) output vector

(18 )

A. M. da Silveira , W. S. Mota and J. C. de Castro

220

TI-!E PARAMETER OPTIMIZATION TECHNIQUE

Consider the parameters of the static compensator represent ed by the vector n =

Table 1 Parameters Data

PARI\METERS

[ ~,

The optimization problem consists in adjusting the parameters n , to Re [ A.1 (A(n) ]

min max

i

1,2, . . n

subjeted to A. (A(n))] 1

I

m Re

A. (A(n))]

S ~

i=l , 2, ... n

1

and ~in

AVAllJlliLl:

OPTIMIZED

!W\GE

nmin 0.1 0. 0 93 0.1 0.253 0.0013

f\n;,,,

(,HE<;!'

It is desired to determine these paramet ers so t hat the model equation (18) is stable and has the highest speed of response without excessive oscil lations for arbitrary initial conditions , subject ed to the damping factor constraint, which should lie within specified bounds .

max

SUPPLIED BY

:;; ni

:;;

~max

i = 1,2 , ... p

where Ai (A(n)) is the i!h eigenvalue of A, n is the number of eigenvalues, ~ is a practical posi tive number corresponding to he desired damping factor, p is the nLoober of adjustable parameters, and nmin and nmax are practical parameter bounds. Since the eigenvalues of A(n) can not arbitrarily be shifed under the Dar~pter adjustment, t he val ue of ~ can not be arbitrary . Our special optimization problem has an objective function of n where the explicit expression of the function is not known. However, by using a computational algorithm, it is possible to know the value of the function for nay value of n. It should be noted that the objective function is not neces sarily differentiable . In this case an optimization technique to minimize the function without calculat ing derivatives seems to be atractive. The direct patter search optimization technique (Hooke and Jeeves, 1962) was chosen due to its adaptation to the problem. The QR method \,as used to compute eigem'alues 9f M n). A penalty- function technique (Gottfried and lieisman, 1973 ) ",as used to convert the constrained problem into an lmconstrained one by defining a new obieti\'e function which includes some penalty for violating a constraint.

0.03 3:6 0.015

GKRC rnRC TMRC

0. 0 0. 43 0. 0035

Table 2 Eigenvalues of the Interconnected

WITH PARM-1ETERS SUPPLIED BY CHESF - 66.6667 + j 0.0 - 33.5537 + j 0. 0 - 2. 0656 + j 11 . 5606 -2. 0656 - j 11 .5 606 -0.4 423 + j 7. 9653 - 0.4423 - j 7. 9653 -1. 0489 + j 8.2449 -1 .0489 - j 8 . 24-19 - 1 . 3011 + j 5 . 9563 -1. 30 11 - j 5. 9563 - 7 .~792 + j 0.0 - 0.2855 + j 0 . 0 - 2.0953 + j 0 .0 -1. 2731 + j 0 . 0

~bdel

WITH PARM-1ETERS OPTHlIZED -50.8 754 + -21.1090 + - 2.0275 + -2.0275 * -0. 5208 + * - 0. 5208 - 0.9856 + - 0.9856 -1.31024+ - 1 . 31024-

- 6. 9749 - 0. 2883 - 2.0586 -1. 2684

+ + + +

j 0. 0 j 0. 0 jl1.1451 jl1.1451 j 7.9912 j 7. 9912 j 8 . 1317 j 8. 1317 j 5. 9479 j 5.9479 j 0. 0 j 0.0 j 0. 0 j 0. 0

One can see that the system with parameters here adjusted is better in the dynamic stability sense due to the position of the dominants eigenvalues (smaller real component) * \vi th respect to its real component . Al though it has been obtained a more stable system with parameters here adjus table, not to much differ ence was observed in the dynamic voltage response Of the static compensator in the original nonlinear system for both set of parameters. HOh'ever, for the cases here investigated, very good dynamic behavior was obtained for the static compensator . See the results of a transient stability program with the static compensator in the original nonlinear model for the follo\,'ing simulations ; 1) Variation of the static compensator reference \'oltage b\' 10 ~ . See in Fig. 8 . its terminal voltage response. 2) Increasing the load at bus 7 bY 100 \1\'..\1' . See in Fig. 9 . the static compensator" terminal \'oltage

response. 3) Three phase to ground fault at hus 21, (nearbv

.-\.PPLIC\TIO\ TO THE E\A.'!PLE SYSIDl

By using the techniques here proposed, the parameters of the static compensator of the example s;'stem are adjusted . Starting from \'alues supplied bv "CHESF", the follo\,ing gains and time constants \,ere obtained . See table (1 ) , assuming E; = 16. To e\'aluate the dynamic stabili n' of the inter~onnected model , i .e. static compensator connected to the dynamic equi\'alent , see the eigem'alues of the system in the following cases : a ) liith parameters supplied by CHESF b) lii th parameters opt imi :ed by the teclmique here proposed. See table 2.

bl~

- ) , lasting 0. 1 s . See in Fig . 10 the static compensator terminal \'oltage response. CO\CLUSIO\S

The techniques here presented ha\'e sho\'11 to be effi cient in the tunning of static compensator parame ters into a large pOh'er s;'stem take into account the d:-namic interaction of the "'hole system . The procedure does not need large computational effort and requires small computer memory . This method may also be used in parameter adjusting and design of others control systems. For all simulations presented, the d;namic response

Parameter Adjusting of a Static Compensator

221

of the static compensator was satisfactory; i.e . fast and damped in agreement with eigenvalues loca tions of the reduced linearized model as requiredby the optimization technique.

original system identified model

REFERE."ICES Brogan, II'.L., (1974 ) . ~1odem Control Theory" (book) Quantum Publishers Inc. Castro, J.C . , (1984 ) . A computational technique for determination of the adjustable parameters of the synchronous machine controller. In Portuguese. Proc. of the 5th Con resso Brasileiro de Automatica 1 ongresso Latlno Amerlcano e Au tomatlca. pp. 667-672. Gottfned, B.S., and J. \\'eisman (1973) Introduction to Optimization Theory. Prentice Hall, EnglewoOd Cliffs. H6oke, R, and T.A Jeeves (1961 ) . Direct searc solution of numerical and statistical problems. J . Assoc . com~ . ~Iach., 8, 212-22 9. ~lota, 111 . 5., (1 81). "optlmal Control Techniques In the Design of Power System Stabilizers", Ph .D. Thesis, University of Waterloo - Canada. - - -

",

0. 5

Flg. 3.

t(s)

1.0

fi iD for variation In

original system identified model

( EOU I VALEl'H

-t>

~

I \"

SYSTBI

\ ~

T

I?

Fig . 1.

lD

v

+

Q

j iQ

Fig. 4 .

VD + j v Q

0. 5

fiiQ for variation

Static compensator/dynamic equivalent system

.

Fi g.

0

..

or

lI'est s'"stem of CHESF (Brasilien Power Company)

It

tlS)

in fivD

1.0

A. M. da Silveira , W. S. Mota and J. C. de Castro

222

V(pu) 1.0

original system identified model

0. 9

o

tls)

0.5

Fig. 5.

6i

D for varIation in

1.0

1.0 0. 5 t (s) Fig. 8. Variation of 10 % in the static compensator terminal VOltage

6VQ

V(pu)

original system identified model

o Fig. 9.

o Fig . 6.

6i

Q

t(s)

0.5 for variation in

t (s) 0.5 1.0 Static compensator terminal voltage for a variation of 100 mVAr on the load of bus 7

V(pu) 1.0

1.0

6v

Q

o

I (pu)

G , R C I"------'----~

Fig. 10.

0.5

J

y

CAP

STABILIZII'.(; SIG\AL

Fig. 7.

t (s)

1. 0

Static compensator terminal VOltage for a three phase to ground fault at bus 21

Static compensator block diagram