COMBINEO Lp (Lı, L2 and Loo) STATE ESTIMATORS IN POWER SYSTEMS

(1)

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COMBINEO Lp (Lı, L2

and

Loo) STATE ESTIMArfORS IN

POWER SYSTEMS

•

Nurten

ERMIŞ

..

Sakarya l1ııiv., Sakarya

MYO.

Elektrik Progranı ı, /

\

cl

a

paza

rı

. Tel:

'6� '77400 1 ( 1 72 ),

n er nı

İs1d

s <ı kar\· a. cd u. tr,

--- ... ·----� __ , ___ _... _______ �

ABSTRAc--r

· ·

In this study. the objccti e is to conıpare the behavior of the Lp (p --

1,

2

and

:x:)

state

cstinıators and thcir conıhinations

"hen applicd to

a

_{three plıase halanced po\ver systenı u nder steady statc conditions. The general approach in} _s

_ta

_te

c"tiınatinıı is ın nıiııinıizc a rcsidual of nıeasurenıent equatioıL. The cstiınatc is dcpeııdeııt on the nornı uscd to nıeasure this ı-c\idual. The _{pcrfnrnıance of the Lp cstinıators is conıpared uııder both nornıal and bad data conditions. The results are}

i llustrntcd us ing the

I

E EE 6, 30

and 5

7

-bu s test systenıs. The canıparison i s co nd uctcd bascd on accuracy and conıputational

ıiınc. rhc ıııain conclusion of this study is that the coınputational tinıe and the nuınber of iterations required for L

1

and

Lx

cstinıators to con\ erge substantially reduced using a canıbination of the

_L2

c tiıııator \Vİth these estinıators. The use of a

"uitl1bk

tiltering technique and \Veighting factors nıay enhancc the estinıator output

"

e�

· " ords -

Po\\ Cr systenı state estinıation. least absolute value

(LA V),

least squares (LS). \Veighted lcast squares

(\\'LS).

ma,iıııuııı absolutc dcviation cstinıators.

• • • • • • • •

(;Uç SISTEMLERINDE BIRLEŞTIRILMIŞ LP (Lı, L2

and

Loo)

DURUM

• • • •

KESTIRIM

CIL E

RI

••

()ZET

Bu �alışınanın anıacı, sUrekli durunıda ve üç t:ı/lı dengeli gi.iç sistcnıl�rine uygulandığında. Lp (p -=

1. 1

ve

x)

dunını kL·.;;ririıncilcrinin ve onların konıbinasyonlarının davranışlarını karşılaştırnıaktır. l)urunı kestirinıinde genel yaklaşını. ölçlirıı dcııklcııılcrindeki f�1rkı nıininıunı yapnıaktır. Kestirinı, bu farkı ölçnıck için kullanılan nornıa bağlıdır. Lp kcstirinıcilerinin pcrfnrınaıısları. hcnı nornıal hcın de kötü veri koşullarında karşılaştırılnııştır. Sonuçlar.

IEEE

6,

30

ve

57

baralı sistenıler kullanılarak elde cdilnıiştir. Karşılaştırnıa, doğruluk ve hesaplanıa süresine göre yapılnııştır. Bu çalışnıanın ana sonucu.

Lı

\C

1

f kcstirinıcilcrinin yakınsanıası için gerekli slire \e iterasyon sayL ı, bu kcstirinıcilerin L--. ke ·tirinıcisi ile kanıbine

edildikleri zaınan azalnıaktadır. Uygun tiltrelenıc ve ağırlıklı faktörler kullanıldığında, kestirİnı sonuçları daha

da

i� ilcştirilcbilir .

.

\nahtar

kelirneler -

Güç sisteınlerinde durtıın kestiriıni, en küçük nıutlak değer, en küçük kareler, ağırlıklı en küçlik karelcr, ıııaksinıum nıutlak değer kestirinıcileri.

•

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to

nıcasurc �ıctı \·c j)O\\Tr,

rc

:

ı�tt\L' pu \\cr.

�

olt�1gc. and currcııts. 'J'hcsc

coııtillunus

or anaing quantitic� �1rc ınonitorcd using

current and

potcntiaı

tı anı..;l'nrnlcr"''. l'hc analog quantities pass

through

t

ra

n"' d

u

c

c r � a n

d

a ıı

a

1

o � u.-to-d i � u. i

ta

ı

c o n v c

rt

cr s,

an

d

the d

i

g

i

ta

ı output is thcıı

tclcınctcrcd

to control ccnters,

over varıous

44 (

'nınhillL'd

ır'

(l.ı

L2

_and

_L

_,

_,)

_t_a

_t

_e _{Estiınators in}_PO\\er

\-.((' .

\.E

cnııın1uniı...;ıtıuıı

_{links. The data received}

_at _the _eııe

�

_{oı_ıtrol} _l'ı..ıi: _rs

_are

_{analyzed and proccssed}

_{by a}_coınpute

ınfurıı1 _tlı�.. _.,_....rcn1 _{operatar about the}

_present

_state

_of

"i)

�tcın. '

ical

nıeasurenıents

_are

_never

_entirel; _free

• 1 llUI�L' atıl

.

\

prcrc, · L'ontrul ' r�1\\ oh'L cfl'cLh ( filtcrınL ı the L·nı nı _ı ad

d

i ti< u ı. s\'stcın -indcpcnd

\'k

(k

ı

\

1l'll'-.lli'L'I tlıc prL''-.l' . �cctıuıı ( 't)Jl�L'lj lll

un

k

nP\\ n rcdund,ıııl :-ı

t

a

t

L'

" t

ı hru:ıd

h

�.1

. .

cstııııatıPı

dctcrı11iııl

the

<lpprı

ı

.

tıınc ı� :ı S\ �tc

nı

disturh<tılL llOrtll(lll\ pruhkın� \\ lllLISt

bt�

LU tılust he

pı

sccnnd".

un1 errors.

n

r

any power systenı sec

u_rity _m_on

_ito

_r_i_ıı_�

...

is a

_{reliable database. In}

_these

_da

_t

_a

_b

_as

_e

_s_·

ı�

_{are systenı ically processed}

_to_attenuate

...

rta

i

nty

of the nıeasurenıent.

_Any

forn1

�

a

loss of information

a

nd

, consequen

terınination of the systeın

state

may

req

..

ureınent

with inherent extra cost.

In

pc

"tate analysis,

for

N

buses, there are 2 .

iables

consisting of voltage

ınagnitu

nd

voltage angi es 8k (k

=

2,

.

..

,N

) at

all

bı.�

l1rc

not

ad

e

qu

a

t

e

for

dynamic

operatio.

ı

n

e

a

s

urcnı

e

nt

errors and possible failure o

· re'll

tinıe data

collection equipme

norc

nıeasurcnıents than

the

nuınbe:

,·

a

r

i

able

s

are

needed.

The

way

in \\hich

t

u

t i

1

i zcd,

g

i

ve

s

ri

se to var

i

o us

techniq u es

)

ıı. S

tat

c

e st

i

nıation techniques ın a:

d

into two

categories, static and

d

y

n

aı

. tic nıethods

nıainly

have application

)r

ıoad tlow

in transnıission

n

etwo

r

k,

''he

ıon of

steady state over a short

per1oc

l�. The

assuınption w

i ll be valid

if

I

ıbjcsted to relatively low

frequen

·\'nanıic

estinıation� on

the

other

hand .

-ı�_a_tc_d

_{with traıısient}

_{or dynaınic}

stab

i

l

the

dynanı i

c of

the

elenıents

of the

sys

..

�red. Rapid data colıection

and

process

.. d,

s

i

n

ce the

ti nı

e sc ale

is

of

the

ord

er

Static �tah.: ,

nation

nıeets alnıost all the

control cen

ııccd" dtıl' \

Lhe

fact that

dynaınic

state

estiınation

coıı1 putatiP n ıl"

i

ntensive.

1 n addition

the

h

ig

unprcdict<lhk

··

n

d

nonlİnear nature of

povver systeıns rnaı

deJiııing .... a n·liabıc nıodel

of

the

dynaınic

power systen

ver \ dil'liL·ult

t

as

k

.

For

a given

povver

system

s

tr

u

ct

u re .. \

ccr

t

niıı

sel

u!' paraıneters

and operating

data, the purpose

tlıc suıtic

c

�

t

i

ın

a

t

io

n is to

c

o

ınp

ut

e

the

network

operatı d�1ta �ı11d

to

aı..;"leınble

a conıplete and reliable

database.

1

unkııo \VIl i

n

d

e

pend

en

t

variables of the estimation

scher

are the

true

ııode

injections

).._)

(1), the

true

structure

g(t)

a

the tr uc

pa

nı

e

ters

p(t). The structure

g(t)

deseri

bes

t

interconnection of power systenı

different

components.

(3)

.

\l !"en Bilimleri Enstitü. ü Dergisi _� 1 O.Cilt. l.Sa_�\·ı '006

rhc

paranıeter \'ector

p(l)

contaıns the line and llleter

(hdractcristics. The unkno\vn dependent state ector

.\-

contains coınplex voltages

V (!}

\Vhich are of dinıension

11 -

�.\"-/.

\vhere

.\'

is the nuınber of systeın buses. The

k

n

n\' n

q

u a nt it i e s a re the nı e as u re ı n c n ts :: (!) c o n s i s t i n g of

tclcınctered line flo\v, bus injections and bus voltages. At a

ll\ed _t_iı

n

_e _{instant the nıcasureınents are related to the}

un

k n o w n var i a b 1 e s as fo ll o w s

::

ll (Y.g.pJ

· e

( ı )

The

nı

c

asurenıent error e is a randonı vector representing

the

normal d ifference bet\veen the actual observation :: and

/1. Bccause of the prcsence of error e, the d inıension rn of

the

ıncasureınent vector:: nıust be larger than the diınension _'-'

n

of

the unktH)\\ n x. A possiblc nıeasureınent of the

redund ane) i s the rat i o

'7

=

1nln.

The error e i s us u all y

cmısed b\ the nıeasurenıent noise, nıodeling errors, � �

conııııunication errors, cading errors and the fınite nıeter

time constants.

3.

Lp E TIMATION NOR!VIS

Rcgrcssion analysis is one of the nıost frequently used

ıııethods in enıpirical science l7]. The description of

collected

or ıneasured data by linear regression functions

\\ith seve ra 1 paranıcıers i s needed fctr ınodel ing and

prcd i ct ing. l n e

I

cctri c power s ysteın app

I

i catio n s,

nıcasurenıents are usually plentiful. The systenı of linear

eq

u

ations i s said to be an over-deternı i ne d systenı. The

eliserete linear

1�1,

approxinıation for the power systcnı over

dctermined set of 1 inear equation is given by

,. =

ll

.r --:

(2)

\\lı ere _r _{i s the res i d u al vector,}

11

_E

Rmx"

_{i s the process}

matrix,: E

lt

i the ıneasureınent vector,

111

is the nunıber

of

ıııcasurenıents and

n

is the nunıber of the unknovvns. The

l_e_n_gt_h_of

r

_i_{n the}

L1,

·

_{nornı i s gi ve n by,}

lll

,.

Ip

�

(

L

,.,

1 J7

)

ı ı> .

(1 <p<oo)

( 3)

1=1

"

In o

r

d

er for x to be the solution the above equation can be

. wrıtten as, "

1/

X

-

:

_ll,,

_ın

i

_n

_ll

J

-

1

x -

::

111,

X

(4)

45

Con1bined

Lp (L1 L�

and

Lr) Statc Estiınator� i

_n_{Pn\\'cr Systcms}

N.Erın iş

.t.

Lp ESTI;VI.ı\.TIO� !\10DELS IN PO\VER SYSTEMS

The sülte of an electrical po\ver network nıay be dcfined as the flow of co nı p lex po,ver through the net \York branches an d the n et i nj e ct i o n of c o nı p

1

e x po\ v cr at e a c h b u s .

T

h c

state of a net\vork at a gi\·en nıonıent nıa) be conıpletel) deseribed by the et of all coınplex b us voltage .

( 5)

It is assunıcd that the connections bet\veen the nodes and the paranıeters of the branches are conıpletely kno\ n. For

a systenı with

.\

buses. by choosing one referencc phase

angle� there are thercfore

7,\'-/

unkno\vn state variables to

be dcterın i ne d i n ord cr to deseri be the s ta te of the systenı. Typically, power systenıs have far nıore nıcasurenıents

ava i

1

ab le than statc variab

ı

es, w h i c h al lo\v the use of

statistical nıethods to esti nıate the s ta te variabtes of the power systeın.

A

nıeasurenıent vector:: nıay be created which contains 111 nıeasurenıents fronı the pO\Ver systcnı. tv1easurenıents

include reaı and rcacti e po\vcr line flo\YS and bus i nj e ct i o n s, v o

ı

ta g c n ı a g n i t_{u d}_e_{s at b u s e s. tap rat i o s for}

trans fornıcrs, and �o nı c tinıes, phasc angi e nıeasurenıents. The

J.\"-1

state variahles const itutc the state vector x, \vhich

is rclated to the nıcasurcnıents � in a regression _{._} nıodel as,

:: =

h(x)

�- e (6)

•

The vector e is the error vector, \Vhich contains 111 variables e1, . . . •

e,11• V

ector e purpose is to account for the uncertainty

in the nıeasurenıents and the nıodel. Hence. e1 . . .

.

. elli are

randonı variables assunıed to have a zero nıean and a known diagonal covariance nıatrix

R,

R

-

diag(rJ/,

.

..

, O'n/).

₍₇₎

The data pattern in :: \Vhich is represented by the vector

h(x)

is fornıulated such that, for an

n1

sets of observations that are related to a nunıber of state variables, x1. x: and

J\v,

through a l i near regression nıodel, are given by,

::

h

_. e .... :n +

h/1111

�-e ' ll

1 (8)

•

(4)

S

,

\

(

'

Fen Bilinıleri Eıısıitüsli Dergisi

1 O.Cilt. I.Sa) ı '006

Equations

(8)

can be \Vritten as,

1 h

h

x +r +e

::1 - 7 1 1.\" 1 : 1X �� _,_ �- n-1· n-1 · n 1 •

i

-

1

... , 171

(9)

The

.\·-/

"ariabi es

h

1,.

h:,

... h11_1.,

are terıned as independent,

or explanator). \ ariabi es, \Vhile ::, is ternıed as the response,

or dependent variable.

r\

data point is

(h,,. h_

... l···hn_,,, .:-,).

and lic- in an \'-diıncıısioııal space. \\lıich is nıade up of t ,,.o s u bspaces. the respoıı sc space and the f�1ctor space a

1

so

called the _{design space.} _{The rcspnııse space ıs oııe}

dinıensinnal and contnins the _{response \t.Hiablc. The fnctor}

�pacc is ınnde up of

n-1.

h-dinıcnsions. The slopes of the l_ı_ypcr_{plane in}

R''

_are _.r1

.

_.

..

,x,·_1,

_\Vhilc _{the iııterccpt is -"}_._\·

He nce, the task is to find x such that the lı: p�r plaııes best

fits the obser \ ;ı tion\) to the independent variablcs. Fquation

(9) C�lll

he represcntcd in

lll<Hrİ\

rorııl a� _.:-

ll\'.

_e. _\\_'_here

ll

hı ı

h2ı

ı

1 .

. .

1 ( ı o)

ı

ııatrı \

ll

i� c n lll� d t h c d c s i g n ın at ri x, an d

x

the paranıeter \ cctlH·.

In

j)U\\Cr

"':>

�tcın� the _{task or state} _C. _{tiınatioıı is,} thus. to use

...,r�lli...,tical ıııelht'tb to liııd .r �uch that the nıeasurenıents

�;,en h\ :: hc">L fit the �110\\ll s: �tcın ıııodcl prü\'İded b: the

�

--Ll.\\ ci�hted Lcast Sqtıitrcs (\\ LS)

("Ji\ cn

the rc�rcssiorı nınde

l

of

(ô),

lct

R

be the _,_·_a_r_iancc

-...

co\ ariance ınatri\. The nbjcctİ\ c function is givcn by,

./

(\)

1

ll/ ,

.

, "l

\' ( )

-�

2

1 ı Oi

1 -'I

,-1 ,. \

,

.

(.:- llx/

N.

1 (

::

//.\)

( ı ı )

,, here

1

n, .' act) as a \Yeiglıt.

\1

ininıizing

./(xJ:

ields

('.l(x)

{) _-

111 f{

₁₍

::

Hx

)

( 1 2)

"

By t�1kiııg the pscudo inversc of

//"

rhe estiınated _x _is

rouııd such that.

( l

11

R ı

I 1

l

:

111 r\-

R:

1

1 .

( 1 ] )

Thi� _{lcads to.}

-

--4G

Combiııcd ıp (Lı

I 2

anel

i ..)

Swtc 1·-.;tinıaıt'r� in Po\\er

\�

_1eı

\.Er

( 1 4)

\Vhere

S

is called th_e hat _{nıatrix, This törnıula} _is _use�

fornı the residual sensitivity nıatrix 11· that _relates

residuals to the errors,

,.

-

"

-- -

�.

-

-_-.. -

(

1 -

S

)

::

- n·-

( 1 5)

'\nte that the _ınatrix

ll'

_{is singular. hcncc there are ınultip}

nJ-\Cctors:: \Yiıich can satisf) r 11'::. lt can

be

seen

th

tlıcsc crror"' are considercd as lincar conıhination�

of

t

residuals.

ı

his i� İnıportant both in

\VL

·

rcsiduaJ anal:

and in c�pl�1ıııing certain propertics of the

�

· L:

_when

gr

crrors are iıı ,ohcd. For e'\anıplc, the propert)

or

ha\ _i

lincar coınhinations _{of large errors rcsulting} _in _srı

rcsiduab. cuıd large error results in � SC\'cral large residuaı .

-t.2.

Lcast .\bsolute \'alu e (LA.\')

The lcast absolu tc _value;

( L.-\ V)

_{eriterinn is based} _on

the

norın and lı;.1s very uscful bad data rcjcction propertic

Equation

(3)

can be solvcd by substituting the _value

of

_r ,.,- for the rcsidual veetar r, \Vhcrc r, and r, reprcsent t

positivc and ncgativc parts of r respectivel).

T

coıı�trainh

ll,

are

1

inearizcd about the current x" which

gı

ri s c to t h c f () ll o '\' i n g, ... ,

.

IJJI/1- • ( L' )

( 1

6) ,. \\ith tlıc con�traiııt ,.

(/.-/.//,)

,.

(ı 7)

X

Equation

( 1 7)

can be solved for

_lr

usiııg tlıe sinıplex

lint

prograınrning ıncthod.

//,

is the Jacobian nıatrix. The\ a of x is thcrı updated, x" 1 = x" ""

L1t.

The constraints

liııcariJcd nt the updated operating point and the procel.

is re peat cd u nt il convergencc occurs.

4.3.

Least squares cstiınator (LS)

rhc kn�t �quarcs solution

(LS)

i· obtained ''he

n

p- 2.

I

(5)

\\C

ren B i l i nıleri

E

nst it

ü

.

ü Dergi i 1

O.Cilt. !.SaYı

2006

� ;

applications. lt provides one \Vay of dea! ing with an over deternı ine d systenı. The LS nıethod is prefen·ed large Iy

because the deternı ination of the d esired paraıneters is ınathenıatically sinıple. Based on the assunıption that the

error r in the nıeasurenıents is independent and normally

distributed, Eq uation

(1)

can be rewritten as,

� _, . .

H

x-

�

l

;

=

(

ll

H

x- z

ll

H

x- z

IIJ

�m zn T T

f! ffy

=

[-[

z

(ı

8)

The q uantity

(H

1 H)-1 H

r is the pseudoinverse of matrix

H. -t-t

'l�"\imurn A.bsolute Deviation

The ıııaxiıııunı absolute value eriterian is based on Loo

norın.

I

n this. the optinıal estinıator nıiniınizes the Hoo

110rtll of the pO\Ver spectruın density nıatrİX of the

cstinıation crror 1'1

such

that,

111

i

n

s

111 CL\

1

ri

1

=

ll

Hx

1 - z 1

!

1

_x

[{

_\ e _z > X

-H

_\ e -7 --( 1 9) (20)

Siıııplcx liııear progranııning nıethod is used to solve for

.:_\x. .\s the constraints are ineq uality constraints, it is

necessary to add both surplus and artifıcial variabtes to each

equatıoıı.

5. SJ!\Il'L:\TION RESULTS

.

·ıııc

1

EEE

6, 30 and 57 bus net\vorks were u sed to evaluate

the coınparison of

L" (L1. L3

and

Lx)

estinıators. The

topologies of the 6-bus, the 30-bus and the 57-bus systeıns

as \Veli as the nıeasurenıent set are shown in Fig.

I,

Fig 2 and Fig 3. Test systenı data are given in reference [8] and

[9] . The nıethods have been progranımed in MA TLAB on a

Pentiuın 1 1 1-800 MHz computer with 5 1 6 MB RAM. The

propo�cd algorithnı was tested on two different cases in

\\lıich the nıeasurenıent sets are assuıned contanıinated with diffcrent levels of no ise. The noise level varies between O%

(case 1 -wlthout noise) to 20% (case 2-with noise) of

test syslcnıl61

THRrE W INC>I lf(l 1 RAIISFORUFR ftllJI\IALt.'flS

tiANCOCK ��NOı<E 30

'{!-

--- 1 • 79 ... CI.O\'E ilC l,Lf @; GEl< ::Rt.lOAS @ SYi'IC.-iROIICUS CO•lGEilSOfiS

Figure 2. IEEE 30-lıus

rest system

ll

OJ.

• . . .. ,.· ,_. /r / 1 • • - . .... .. / 1 - -••• 1 .. ... � , • •

9

. -...-. ... · .. i ..C _ ... ;o 1;' -( --(1 ---... _ l -... 1 ( ( ! ' ( / , � � • � •• 1 • ) . � .. � --·-,.� · ... ' . . .·t-i . , . .., ... : . - · -• ... .... - -... �- . , . ... .. ₇'"''\ ,* , _.._._._·_.. �· ·'.s' •/ ... •. ' (/, ; �.' � .. _.;:;....;:;...:...- ... ' -- ' ·· -...r ... "', ., ...:.. . ( ... . ' _:--•ı• · .. �. . . � ... : ·- ---- ----__ : ·- _{··-- .}.;...;.. - t _: 5 ··- � .. !.': ... --

(6)

SAU Fen Bilimleri Enstitüsü Dergisi 1 O.CiiL

ı

.Sayı 2006

Read ıneasurements Rcad system torolog)

•

Filtering the input uata

Construct Ybus ınatri\

eta starting value of

the statc \'ariablc

Calculatc the misına.tch

M

ismatch < 0.00 ı o Construct .J acobian 111:11 ri\" f f

,..---�':...___---.

Ruıı the l.11

statc cstimator. solvc ror

Lh

f-f L1X

=-: lJpdatcx x'k IJ

-

_/kJ

+ L\x Yes .... Outrut to

o

th

e

r program s

Fig u n .ı. Po\,·cr ll n w st u te e ·ti mu ti

o

n

[ 1

1

ı.

The output ro r e a c h ca s e i n c lu d e s the b u s v o

1

ta ge nuı g n it u d e s a n d

,·oltagc phase angle . The residuals for both t\VO cascs are

ca \culatcd and p lo tt cu. The com pu tat i ona 1 ti ın e co n su med by ca ch

statc cstimator i� ıııensureu and conıpared. Table l . Table? and

l abk

�

contain the output results for ,,·ithout and ,,·ith noise cases

undcr study. lt can be

i

ni'erred

fron1 Tablc 1. 2

and 3 that the

ıncthod of the least ab

so

lute value e.ives the least residual and '-"

hence the more accurate re� u lt. The co ın putaticnal time is the least

in ca �e of the Ica t ·quare e. tinıator.In lo,,· no ise condition. th

e

48

Coınbined Lp (L1 L2 and LYJ State

E

stimato

r

s in Po\\ er

.'�stern

\.Erm

tvvo ınethods give _{equally accurate results. But} _with _high _no data. \vhich is the _{practical case in power} _application. _ı least absolute value gives better results.

Using a canıbination of the

L'}

estinıator w

_i

th the othr

estinıators can substantially reduce the conıputational tim

and the nuınber of iteration needed for L1 and

L7•

In

th

conıbined ınethod, _{the least square estinıator}

L:;

_is_execute

for one iteration, and the n the other esti nıator

is

applie

For

IEEE

6-bus, _{30-bus and 57-bus test systenıs} _with

_the

saıne input data, when this new cornbination method

applied, the results in Tab le 1 , Tab le

₂

and Table

3

a

obtained.

Ta b le 1. Co ın parı'-ıon wıth respect to resıduaL and computntıoııal tıme

Lp estiıııator� t(x the 6-hus test S) stern

Rcsidual

s

Coınputational Time

(see.)

LP

ı

norm

V..' ithout

W ith

W ithout

�'ith

No ise No ise No ise o ı se

L, '). 7804 2.7837 0.93 0.9-l

L,

-

!

.7X08

2.7850 0.0600 0.0500

-

-/,,

1.7X3?

2.7S78 0.3800 0.3900

-

·--'-· · ·

Tahle 2. Cnınpmı�nıı \\ith rcspcct to resıthıals nncl coınputatıonal tım'"

Lp estı ın�ttl)t � rur the 10-bus test system

1 C

ompu

t

atio_{nal Tiıne}

Rcsiduals

(se

e

.

)

LP

norm

_{\V ithout}

W ith

W ithout

\V ith

No ise No ise No ise No

ise

1�

1 13.4091

1 3.4105

2.600

4.0 ı o

L, 13.4416

13.4385

0.600 0.610

Ly 13.-l227

13.4155

5.930 6.150

Tabi� 3. Cnınparısnn \\ ıth rcspect to residuals and coınputationul tı

Lp estiınators li.)r th

e

)7-bus test system

�-Resi d u al s LP

norm

_W

ı

1

ı

•

(7)

� \ c·

ı- .'" - ı ... ı B ı- ı ı· ı ... , ıı ._'"_{- ı-,}·

l

-

_{n. ı \lt h}u D�ı _�""'

\ lLl

_ı

\ı. \

_"-ı�l\ı _"1()06

....

l�ab\c -+ shl")\\ S

t

ha

t

th�

(0tı1ptıtational tinıe i nıuch

ın

o

r

e

l e

s s

than it· � ne�dcd

itH

!�1

,\nd 1

��ti nıators. indi\

idu

all)

.

l t i s

ad,

i �a b l c tı._) u�c the ccıınhincd

!...,,

nıethod lO estimate

t h e ' 1.)

l t age atH!lt.!-... and

� ' olta�� ınaL?.n itudcs i n buses. _�

-r

-

--- --- -- -ı . ı

ı

_. _-

1.:-r

Rc-.ıdıı.ı ı ...

r--T

___

T

1

� ,... ,_ :J -1 ·..r: -J ı ...--- -- h-ı _bth

0�3 1.3-W - - - . T :.ı ./ -,.. -z ,-- -:S 0.1 J () � 0260 .2 .-ı_-o - - - ---- - - · - .. J 0ı) - ----Case No

Fig u r·e -t. C\)ın par ıson \\ ıth resreel to conıputat ıonal tiıııe ol' the L , and

llıL' u ıııb ıncJ L, ( ! _ _, , L . ) ustmwtors tor the 6-bus test system

G) E ı-ro c o -ct. ::ı o. � · -o o • 4G ı :?0 · oo ₁ 0 80 . O GO . D ..:Q . o 20 1 . - --+- L --.- L _--L -- • -- -- -�--- ---· 0 0-J --- ---, 2 Case No

Fi�urr 5. C\mırarıson w ı t h respcct to Cl)ınputat ınnal tımc of the 1.1 and

thL' C l ' ıııhııı�d L,. ( L_,+L1) esrimators for the 6-bus test s :ı stcm

49

Coınbined Lp ( Lı

l.� and L , )

Statc Estimators in f\)\\·er Systeırıs

N .

Er nı iş

F i gure �

-L

6 an

d 7

shO\\

c

a

nı

parison \Vith respect to

c

oı

npu

t

at ion

a

l tinıe of the L >' and the conıbined

1

.. 1,

( /.�- !�

/ )

estinıator.

·

for

the 6-bus test ) stenı , the � O-b us test sy stenı

and the 5 7-bus test systenı respect i\'el) . Figure 5 shO\\'S canıparison \\'ith respcct to

c

onı p ut

at

ion

a

l ti nıe of the L 1 and the con

ı

b

i

n

cd

1.1'

(

IJ:

+-

t

1J

esi

inıutors

for the 6-bus test

S\ stenı . .. .... , • • •· ı ' • 6 )J. C> f "> ₁ • iij c o ;; "' _.:_{� .} ;;ı a. E o u j • • ı 00

1

2 Case No

Figurr 6. Coınrarısnn '' ılh rcspect to comrutatıoııal tınıe of the /. , and the cnıııbıncd L, ( /._,+ /. ,) estimators ror the 30-bus test s� steın

ı 4 00 • -+- ı . .... ı -1. 1 :? 00

l----

---+ :ooo tU E 1-ııı a oo ' c Q -ro :ı 600 . o. F o u _.ı₀₀ _. :ı ()(o . -000 -- ---Case No -· 2

Fig u rc 7. Comparison with respecl to coın p u tational time of the L , aııJ the combiııed L1ı ( L.ı+ /, , ) estimators kır the 5 7-bus test system

6. CON LUSIONS

Th is paper presented a new and effic ient conıbi ned nıethod to esti ınate the vol tage ( ang i es and nıagn it udes ) i n systenı

b uses. The proposed nıethod based on canıbi nation

!.fl

est i nıators for enhancenıent of com p utatİona I _{speed . . A s the}

proposed nıethod converges nı u c h faster than the least absolute value esti ıııator fo r l arge power systenıs, the

conıputat ional t i nı e and iterat i o n n u nıbers needed

for

convergence substantiaJ iy red uces. At the sanıe

ti nıe.

r c s i d u a

1

s o b ta i n e d u s i n g a c o ın b i n a tion of

L

1 J n o

rııı

s ar e t h c

sanıe as residuals obta ined u s i ng L1)

norıııs

individua l .

(8)

S/\Cı

fen _{B i l im leri E nstitüsü Dergisi ı O.C ilt.} ı .Sayı

?006

cstiınation technique based on L,. norms is a vcry c f'fcctivc tool

and can be u

pp

l i ed i n a \·ari et) of are as i n engineering.

.. \ C K N O\\' L E DG '1 E N TS:

J \\'O tı l d like to express our deep thanks and heartfelt

2:ratitude to the excellent assistance of Dr. !Vlaher Bakroun

�

i n gathering soıne of data in po,ver engineering departınent at Arizona State L ' niversity.

R E F E R E N C E S

f 1 ] . C . Rakpentlıai � S. Prenı rudeepreechacharıı. S . Uatrongjit. N . R . Watson,� '' Measurenıent placenıent for power system state estinıation using decoınposition

techn iq u e''. Electric Power Systenıs Research, 7

S, .ı

1

-49. 2005.

[ ?] . N . Ernıis,

H .

Cakir, "Guc Sisteınlerinde G urultu l u

D urun1da Lp DurLıın Kestirinıcilerin i n

K arsilastiri l nıasi''. S A U Fen bili nıleri Dergisi, 7.Cilt,

1

. Sav _"' ı � Mart ' O O 3 .

[ 3 ] . C . Rakpenthai, S. Prenırudeepreechacharn, S .

Uatrongjit N . R . \Vatson,, ''M casurenıent placeınent for povver systenı state estinıation using decoınposition technique". E l ectric Power systenıs Research, 75, ..ı l

-+9. 1005.

so

Coınbined L r ( L 1 L2 and L ,. ) S tate Estimator i n Po\\er ' ste •

N.Erm

[4]. M . B . Do Coutto F i l ho, A .

\1 .

Leite da Sil a and

o

Falcao, " B ib l iography on Power System 1

Esti n1ation ( 1 968- 1 989

):

'

I

EEE

Transact

i

ons on Po

S

y

st e nı s, 5 -3 , 9 5 O-9 6

I .

1 9 9 O .

f S ] .

_G.

T. Heydt. _{··conıp uter A nalysis} _Meth_o_ds

_f

_or

_p0,

Systenıs," Stars _{i n a Circle Publications,} _Sc0ttsd

AZ, 1 996.

[ 6 ] .

D. M .

_{Fal cao and S.}

_{M .}

_De _{Ass is.} _{· Line}

Prograınnıing State _{Estinıation : Error} _Anahsis _a

Gross Error I dentifıcation,'' f EEE Transac

t

ions

Povver Systeıns 3-3 . 809-8 l S , 1 98 8 .

[7] . Abur and M . K . Celik, "A Fast A lgorithnı the \\ ei{!h _""

Least Abs o I u te

V

al u e S tat e Esti mation:'

_lEE

Transactions on Power Systenıs,

6- I , 1 -6� 1 99 1 .

[ 8 ] . Spath.

H ,

�'Mathenıatical A l gorith ms for Lin

Regression," Acadenıic Press, U K, 1 987.

[9] . M. Moghavven1i and

_O.

Faruque, ··Real-tı

Contingency Evaluation A nd Rank ing Techniq

Generation, Transn1ission A nd Distribution." lE

Proceedi ngs, 1 45-5,

S

1 7 -S24, 1 99 8 .