Mili_Bad Data Identification Methods in PS State Estimation_A Comparative Study

8/20/2019 Mili_Bad Data Identification Methods in PS State Estimation_A Comparative Study

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I E E E

T r a n s a c t i o n s

o n

P o w e r

A p p a r a t u s

a n d

S y s t e m s ,

V o l .

P A S - 1 0 4 ,

N o .

1 1 ,

N o v e m b e r

1 9 8 5

B A D

D A T A

I D E N T I F I C A T I O N

M E T H O D S

I N P O W E R

S Y S T E M

S T A T E

E S T I M A T I O N

-

A

C O M P A R A T I V E

S T U D Y

T h .

V a n

C u t s e m

M . R i b b e n s - P a v e l l a

D e p a r t m e n t

o f E l e c t r i c a l

E n g i n e e r i n g ;

U n i v e r s i t y

o f L i e g e

S a r t - T i l m a n ,

B - 4 0 0 0 L i e g e ,

B e l g i u m

Abstract

The

identification

techniques

available

today

are

first classified

into three broad classes. Their

behaviour

with

respect to

selected

criteria

are then

explored

and

assessed.

Further,

a series

of simulations

are carried

outwith

various

types

of

bad data.

Investig-

ating th e

way

these

identification

techniques

be -

h av e a ll ow s

completing

and

validating

th e theoretical

comparisons

and

conclusions.

l a

I N T R O D U C T I O N

In

the

list

of a power

system

s t a t e

e3timator

soft-

ware

routines, bad

data

identification

is

th e

last

but

not

least

-

satellite

function.

Its task is to

guarantee

th e reliability

of

the data

base

generated

through th e

estimator.

I n d e e d ,

despitethe

preprocessing

data validation

techniques

used

to

clear the

d a t a

re-

ceived at

a

control

center,

gross

anomalies

(suchas

bad

data,

modelling

and

parameter

e r r o r s )

may

still exist

d u ri n g e sti matio n.

To avoid

corrupting

the

resulting

data

base, it

i s

of great

importance

t ha t t he se

anoma-

lies

are

identified

and

further

eliminated

from

the

set

of

measurements.

This explains

why

t h e need for

a

func-

tion

capable

to

identify ba d

data has

been

felt almost

simultaneously

with th e need

for

the

state

estimation

function

i t s e l f .

It

also explains

the number

and

diver-

sity

of

research works

carried out

on

th e

subject.

This paper

aims

at providing a

comparative

assess-

ment

of

th e post-estimation

identification

methods(1)

available

today.

More specifically

it concentrates

on

evaluating

th e

techniques

able to identify

bad

data

( B D ) ,

i . e . grossly

erroneous

measurements.

T h e s e

tech-

niques

are

first

classified,

then

explored and

compared.

Three

broad

classes are

distinguished

:

the class of

i d e n t i f i c a t i o n

b y

e l i m r r n a t i o n

( I B E )

( 1 - 1 4 ] ,

that of

the

n o n - q u a d r a t i c c r i t e r i a

( N Q C )

[ 3 , 1 5 - 2 0 ] ,

and

th e

h y p o -

t h e s i s

t e s t i n g

i d e n t i f i c a t i o n

( H T I ) [ 2 1 1 .

The

investi-

gations

are

based

upon

both theoretical

considerations

and

practical

experience.

Th e

latter

has been

acquired

through

simulations performed

on

four different

power

systems.

T he

results

reported

here

concern

simulations

performed

on

the IEEE

30-bus

system,

with th e

three

possible

types

of multiple

BD

:

noninteracting,

inter-

acting,

an d

unidentifiable

ones.

The

paper

is

organized

as

follows.

Section

2

gathers

th e

material

necessary fo r

th e

intended

explo-

ration.

The

reader

is

supposed

to be familiar at

least

with

state

estimation

and BD

detection

techniques;

s o

this

Section

focuses essentially

on

topological

identi-

fiability

aspects

and

selection

of identifiability

criteria.

Section

3

gives

a

brief description

of

the

various

identification

methods within

their

correspond-

in g

categories,while

Section

4

investigates further

and

compares

th e three

main

methodologies.

Finally,

th e

ex-

ploration

is completed

an d validated

through

the simu-

lation

results

of

Section

5 .

85

WM

060-9

A

paper

r e c o m m e n d e d

a n d

approved

by

the

IEEE

Po w er

System

Engineering

C o m m i t t e e

o f

the

I EEE

P o w e r

Engineering

Society

f or

presentation

at th e

IEEE/PES

1985

Winter

Meeting,

New

York,

New

York,

February

3

-

8 ,

19 8 5 .

Manuscript

submitted

January

1 9 ,

1 9 8 4 ;

made

available

f o r

printing

November 1 9 ,

1984.

2 ,

MISCELLANIES

Somewhat

hybrid,

this S e c t i o n

groups

the

various

pieces

of

information n e c e s s a r y

for

t h e

subsequent

developments.

The

degree

of

the

authors'

personal

per-

ception

and interpretation

g oes

increasing

along

the

paragraphs.

Starting

w i t h

d e f i n i t i o n s

o f

t h e

usual

symbols

in

§ 2 . 1 ,

one is

l ed

up

t o

some

u s e f u l

topo-

logical

considerations

an d definitions

in

§

2.3 and

2.4

and finally

to

the

selection

of

relevant identifiability

criteria

to

be used

in the

comparative

a s s e s s m e n t

of

the

various

identification

methodologies.

2.1. STATE

E ST IM ATIO N:

DEFINITIONS

AN

SYMBOLS

N. B .

W i t h o m e

o b v i o u z

e x c e p t i o n 2 ,

f o w e L

c a 6 e

i t a & & c

Z e t t e A 6

i n d i c a t e

v e c t o 4 ,

c z a p i t a

ittc

a n d

c a p i t a t

G ' t e e k

t e t t e A

d e n o t e

m a t i c e a .

On e

s e e k s

the

e s t i m a t e

I

of

the

true state

x

which

best

fits

the

measurements

z

r e l a t e d

to

x

through

the

model :

a

4

_

)

where th e

customary

notation

is used:

z :

th e

m-dimensional

measurement

vector;

x

:

the

n-dimensional

state

vector

o f

voltage m a g-

nitudes

and

phase

angles;

n

=

2 N-

1 ,

N

being

the

nu m b e r o f

system

nodes;

e

the

m-dimensional

m e a s u r e m e n t

error

vector;

it s

i-th component

i s 2 :

a normal

noise

N(0,Ui)

if

t h e

corresponding

measurement

i s

valid,

-

an

unknown

quantity

otherwise.

Moreover

use will

be

made

of

the

variable

ei

v i -

E [ e i J ,

w h e r e

E

stands

f o r

expectation.

The

weighted

least

s q u a r e s

( W L S )

e s t i m a t e

satisfies

the

optimality

condition

H T ( . 1 )

R- [z-

h ( ± ) I

=

H T ( e )

R

1

r

=

0

( 2 )

where

H A

ah/ax

denotes

the

Jacobian

matrix,

R =

d i a g ( a t )

an d

the

measurement

residual

vector

i s

b y

definition

.

A

where

W=

IBEHTR

1

an d

E -

( H T R - l H ) 1

( 3 ' )

In

th e

absence

of

B D ,

the

measurement

residual

vector

i s

distributed

:

N ( 0 , W R W T T ) =

N ( O , W R )

The

presence

of

BD

i s currently

detected

through

one

of

th e

variables below

:

the

weighted

residual vector

rW

R - r r

(4 )

th e

normalized

residual

vector

rN

=

r

with

D

iag WR)

5

-

th e

quadratic

cost

function

J ( e )

=

r T R - U

r

=rWrW

( 6 )

2.2.

DETECTABILITY

OF BA D

DATA

F or any

detection test,

th e probability

6

of non-

detecting

BD

i s given

by

6

=

prob

(ji|


2/13

3 0 3 8

where

; i s

th e

statistical

variable of c o n c e r n

( r w i ,

rNi

o r

J

)

with meanvalue

1 p

and

variance

C 2

;

X i s

the

detection

threshold.

Hence,detecting

the

presence

of

BD

requires

that

[ 2 4 1

| 1 |

>

X-N

N 7 )

Let us

now

consider th e

case of a

s i n g l e

B D .

D e f i n i t i o n .

Given

a n

error

probability B

t h e

d e t e c t a -

b - i Z l i t y t h r e s h o l d

o f t h e

i-th

measurementis

defined

as

t h e

minimal

m a g n i t u d e

of th e

corresponding

w e i g h t e d

error

e

necessary

to

detect th e

presence

of

BD with

a

probability

Pd =

1-S

of success

( t h e

other

measurements

being affected b y g a u s s i a n

n o i s e s ) .

F i g .

1

shows

th e

value

of

th e

relative

d e t e c t a b i l i t y

threshold

corresponding

to

th e

rw, rN

and

J

tests

as

a function

of th e

W i i

coefficient.

These

curves,

p l o t -

ted

via e q .

( 7 )

, -

i n s p i r e

th e

following

comments

( i )

i n

presence

of

a

s i n g l e

BD

( a n d

in th e absence

of

c r i t i c a l p a i r s

[ 2 1 ] ) ,

t h e most

p o w e r f u l

test i s

the

on e based

on

r N ;

recall moreover

that within

th e

linear-

ized

approximation

an d

p r o v i d e d

that

e j

=

0

( V j

i ) ,

the

largest

normalized residual,

I r N i l m a x ,

c o r r e s p o n d s

to

th e

erroneous

measurement.

This

i s

g e n e r a l l y

not

true

fo r

I r W .

I m a x

*

Hence

th e

advantage

of

r e l y i n g

on

normal-

ized

r a t h I r than

on

w e i g h t e d r e s i d u a l s ;

( i i ) when th e

local

redundancy

d e c r e a s e s ,

W i i

decreases

t o o ; ,

h e n c e ,

in order

to

be

d e t e c t a b l e ,

t h e

e r r o r s

m us t b e

l a r g e r ;

( i i i )

critical

measurements

a re characterized

b y

W i i =

0:

their

errors

are

thus

undetectable.

Indeed such measure-

ments

have

a l w a y s

n u l l

r e s i d u a l s ;

( i v )

i n

t h e

p r e s e n c e

o f

r i u l t i p l e B D , p r o p e r t y ( i )

d o e s

not

hold

anymore. I n d e e d ,

i n

this

case,

E [ r N i ]

i s

a

linear combination

of

the

gross

e r r o r s

( e . g .

s ee

( 2 . 1 1 )

in

[ 2 1 1 ) ;

( v ) d e s p i t e

th e

above risk

o f erroneous

j u d g e m e n t ,

th e

rN

criterion

s t i l l

remains t he m o s t reliable one;

i t will

t h e r e f o r e

b e

u s e d

t o

determine t h e

s u s p e c t e d

m e a s u r e -

ments :

these ar e

measurements

p o s s e s s i n g

normalized

residuals

larger

than

the fixed threshold.

2 . 3 . TOPOLOGICAL

IDENTIFIABILITY

OF

BA D DAT A

Given

a se t of

BD

i t i s

i n t e r e s t i n g

to

determine

whether

th e measurement

c o n f i g u r a t i o n

is

rich

e n o u g h

to

allow

their

proper

identification.

D e f i n i t i o n .

A s e t o f

B D i s

s a i d

t o be

t o p o Z o g i c a Z Z y

i d e n t i f i a b l e

i f t h e i r

s u p p r e s s i o n

does

n ot c aus e

-

s y s t e m ' s

u n o b s e r v a b i l i t y ,

-

creation

of

critical

measurements.

P r o p o s i t i o n .

To b e

identifiable

a

set

of

BD

must

neces-

s a r i l y

be

t o p o l o g i c a l l y

identifiable.

This

proposition expresses

th e

f o l l o w i n g

evidence

in order

to

i d e n t i f y

f

BD

among

m'

measurements,

i t i s

necessarythat

f

<

m ' - n '

,

where

n ' i s

t h e number

of unknows

to be estimated.

Note

t ha t t hi s

i s a

necessary

but not

suffi-

cient

condition

fo r

proper

i d e n t i f i c a t i o n ;

indeed numer-

i c a l

aspects

have

also

to

b e taken into

account.

A reliable

identification

p r o c e d u r e

should

b e

able

to

recognize topologically

unidentifiable

B D ;

i n such

cases, i t s ho ul d d ec la r e the

problem

unsolvable

an d w a r n

th e

operator

a g a i n s t

th e

lackof

reliabilityof

th e

avail-

able

state

e s t i m a t e ,

rather

than

g i v e

unusable r e s u l t s .

2 . 4 .

MEASUREMENTS

B E CO MI N G C R IT I CA L

D U R I N G

ELIMINATION

Id en tif i catio n me tho d s

based

on

( s u c c e s s i v e )

elimin-

ations

of

measurements

may

lead

to situations

where t h e

remaining

measurements

a re critical : the detection

tests

a r e then

n e g a t i v e ,

since

e rr or s o n critical measurements

a r e undetectable. Now

i t

i s

p o s s i b Z e

that e r r o r s

remain

o n these critical

measurements,

which

would

h e a v i l y

affect

th e

accuracy

of

the final

state estimate

( t h e

r e -

maining

errors

being

no

l o n g e r f i l t e r e d ) .

In

such

c a s e s

neither

of

th e first two

o b j e c t i v e s

of

§

2 . 5 i s attained.

Note

that

n e w

critical

measurements

may

be

gener-

a te d b ec aus e

o f :

th e

presence

of

t o p o l o g i c a l l y

unidentifiable

BD,

-

the

undue

e li mi na ti on o f valid

measurements.

w

-

a i s

th e

false

°

alarm

p r o b a b i l i t y

0 0 2

0 . 4 0 . 6 0 . 8 1 . 0

F i g . 1 :

D e t e c t a b i l i t y

t h r e s h o Z d s

v s .

W i j

In

order to enhance the

r e l i a b i l i t y

of

the

final

data

base,

we

propose

th e

f o l l o w i n g p o s t - e l i m i n a t i o n

procedure

:

( i )

search fo r

al l

measurements

become

critical

after

elimination;

( i i )

ad d

these critical measurements

to the list

of

th e

measurements

declared

f a l s e ;

( i i i )

determine

th e

estimates which

would

be

affectedby

possible

errors

on th e

critical measurements

and

j o i n

this

qualitative

information to the

f in al d at a

base.

Step

( i )

c an be

carried out

by s i m p l y

c o m p a r i n g t h e

lists of critical measurements

b efor e a nd

after

elimina-

t i o n .

The

above

procedure may apply

to

any

identification

m et ho d w hi ch i nv ol ve s

elimination

o f m e a su re m e nt s .

2 . 5 .

P E R F O R M A N C E

A S S ES S M E N T C R I T E R IA

F iv e c ri te ri a

are

selected f o r

a s s e s s i n g t h e q u a l i t y

of th e v ario us i d en tif i cati on

methods.

Th e

first

three

of them

ar e

th e

main

objectives s o u g h t by any

identifi-

cation

approach

as

s u c h .

Th e

two others concernits

prac-

tical

f e a s i b i l i t y ,

i . e . th e

a p p l i c a b i l i t y

r e q u i r e m e n t s .

L o c a c U z a t i o n

o i

t h e

B V

:

a b i l i t y

t o

l o c a l i z e e xa c t -

l y

th e

BD,

or

at least

to

furnish

a list

of

suspected

measurements

which includes

al l

the

BD

and

as

fe w

a s

possible

valid data.

C o t e c t i o n

o J _ t h e _ n a . t

d a t a o bae

: t h e

a p t i t u d e

fo r

c l e a r i n g

th e final d at a b as e

i s

of

great p r a c t i c a l

i m p o r t a n c e

and one of

th e most

essential

tasks

of t h e

overall state

estimation

process.

R e o S o n i i o

o _ p a g e g c a y _ u n d e n t i e a b t e

BD:

whenever

such

BD

a r i s e ,

th e

a l g o r i t h m

should

be

able

to

draw

up

an

as reduced as

possible

list

of

suspected

mea-

surements while

containing

al l th e

B D ;

moreoverit

should

w a r n th e

operator

of

it s

unability

to

i d e n t i f y

the sus-

pected

data

which

have

become

critical and

t h e r e b y

th e

p o s s i b l e

existence

o f

e r r o n e ou s estimates

rather

than

provide

him with

misleading

results.

T m p t e m e n t a t i o n _ & e q y W L e m i e n t s

:

p r a c t i c a l

consider-

ations

o n

th e

i m p l e m e n t a t i o n

and

d e s i g n

should

be

taken

into

account,

such

as

s i m p l i c i t y ,

a d a p t a b i l i t y t o

s y s t e m

m o d i f i c a t i o n s ;

to

a

lesser

e x t e n t ,

memory

s t o r a g e .

C o m p y , t e x t

Ve

i t

s h o u l d b e a s short

as

p o s s i b l e

so

as

to

c o m p l y

with th e real-time

r e q u i r e m e n t s

of

th e

overall

o p e r a t i o n .

3 . BAD DATA

I D E N T I F I C A T I O N .

BRIEF

OVERVIEW

Tw o criteria

a r e

used

to

c l a s s i f y

the various

BD

i d en tif i catio n me thod s

:

-

the nature of th e statistical

tests

of concern,

deter-

mined

b y

the variables

t h e y i m p l y ,

-

the

way

of

e l i m i n a t i n g

BD an d

c l e a r i n g

the data

base.

The first criterion

leads to

d i s t i n g u i s h

H TI

from th e

other

m e t h o d s ,

whereas the

second

leads to

r e g r o u p i n g

th e

various

nonquadratic

criteria

in

a

class

distinct

from

that of

the

elimination

p r o c e d u r e s .

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3/13

3 . 1 .

IDENTIFICATION

BY ELIMINATION

( I B E )

Conceptually,

this

identification i s

th e continua-

t i on of t h eB D

d et ec ti on s te p

which

i s

a

global

criterion

implying

th e

residual vector

r . T he

leading

idea i s

that

in

th e event of

a

positive

detection

test,

a

first

list of

candidate BD

i s

drawn

up

on

th e

basis of an rN

( o r

r W

test,

t he n s uc ce ss iv e

cycles

of elimination-

reestimation-redetection are

performed

until the

detec-

tion test

becomes

negative.

Two

subclasses

may be

distinguished corresponding

to th e elimination of

single

or of

grouped

BD .

I ntro-

duced

by

Schweppe et a l .

[ 1 ]

almost

at the

same

time

with th e state estimation

i t s e l f ,

the

f or me r c on s is t s

in

eliminating at

each

cycle th e

measurement

having

the

largest magnitude of the

normalized or

weighted

residu-

a l .

As fo r

th e

grouped elimination,

a

grouped

residual

s ea rc h h as been

p ro po se d b y

Handschin

et a l .

[ 3 1 ;

it

consists i n eliminating a group of suspected measure-

m en ts w hi ch s up po se dl y i nc lu de s

al l

B D , and

reinserting

them afterwords

one-by-one.

A no th er v ar ia nt

of these procedures

consists

in

solving eqs.

( 3 )

with

respect to on e or several suspected

measurement

e rr ors , t he n

in

c or r ec t in g t h em b y substract-

ing

t he s e e rr or s.

This

measurement error estimation

has

f ir st b ee n proposed b y A bo yt es and

Cory

[ 6 ] .

Later

o n ,

Garcia

et

a l .

[ 7 , 8 ]

have

explored

th e

simplified

way

o f

correcting

one measurement

at

a

time ( t h e one having at

each

step

the largest

I r N i l

)

a nd k ee pi ng the

W

matrix

constant

during

the

subsequent computations of

r N .

Note

that this

technique

has also been

a pp li ed b y Simoes-

Costa

et

a l . [ 1 4 ]

to

the

orthogonal row processing

sequential

estimator.

The work

b y Xi ang

Nian-de

e t a l .

[ 9 - 1 1 ]

has significantly

contributed to e l u c i d a t e

this

question.

These a ut ho rs h av e

brought up

the

singular

character of W , have proposed

it s partitioning so as

to estimate

only

s

( s <

m - n )

out of th e

m

measurement

errors.

Moreover,

they

have

clearly pointed outthe fact

t h a t

c o r r e c t i n g t h e s e

s

m e a s u r e m e n t s

a m o w n t s t o e Z i m i n -

a t i n g

t h e m .

A t t e m p t i n g

t o

i m p r o v e

t h i s

t e c h n i q u e ,

MaZhi-

q ui ang p roposed to

process

combinatorial

sets

of

sus-

pected

measurements

and

to

identify

th e

BD

through a

detection test

based

on an

interesting

formula

he e st ab -

lished

in R e f .

[ 1 2 1

( s e e

§

4 . 1 .

3

b e l o w ) .

Now,

becauseof

th e

equivalence

between

correction

and

elimination,

th e

fact

remains

that a ll th es e

techniques

belong

to

the

class of

the

procedures by

elimination.

3 . 2 .

N ON QU AD RA TI C CR I TE RI A ( N Q C )

Almost in parallel

with th e a b ov e a pp ro ac h,

theNQC

h av e s ta rt ed

b e ing de ve loped

and

explored.

The

idea of

this

m e t h o d o l o g y

differs

t o t a l l y

from th e

preceding

o n e :

here th e

identification-elimination of BD i s

part of th e

state

estimation

itself.

Th e

r e j e c t i o n

of

the

suspected

measurements

depends

upon

the

magnitudes

of th e

( n o r m a l -

ized or

w e i g h t e d )

residuals :

th e

larger

th e

residual,

th e

smaller

th e

weight

allocated to

th e

corresponding

measurement, an d th e

larger th e degree of it s

r e j e c t i o n .

Initiated

by Me rr il a nd S ch we ppe

[ 1 5 1

th e NQ C meth-

ods have been

further

developed

and

analyzed by Handschin

et

a l . [ 3 ]

and

by

Muller

[ 1 7 1 .

More

r e c e n t l y ,

a

compara-

tive

study

of some of

them

has

been carried

out

by

Lo et

a l . [ 1 9 1

an d

by

Falcao et a l . [ 2 0 ] .

3 . 3 .

HYPOTHESIS T E S T I N G IDENTI FICATION ( H T I )

Unlike the

two

previous methodologies,

H TI uses in -

dividual

c r i t e r i a , particularized

to

each

suspected

mea-

surement.

The variables of concern here

are th e error

estimates,

e s

,

of

some

of

th e

suspected measurements;

these

are

evaluated

through

a

suitable

partitioning

of

e q . ( 3 )

an d

a

linear

estimation. Exploiting

th e s ta ti s-

tical

properties

of

each e

through

an individual

identification

testing

allows

deciding

whether

the cor-

responding measurement

i s

erroneous

or not.

T hi s m et ho d

along

with tw o s tr at eg ie s f or t ak in g d ec i si on s

i s devel-

oped

i n

R e f . [ 2 1 ] .

3 0 3 9

4 .

BA D

DATA

IDENTIFICATION. C R I T I C A L . ANALYSIS

4.1.

IDENTIFICATION

BY

ELIMINATION

( I B E )

4.1.1.

Description

The

me th od s o f

this class re ly

on

th e

r W

or the

rN test.

The

c ho ic e b et we en

r W

and rN

implies a

trade-

off

between good applicability features

(simplicity,time

an d c or e saving s)

and

reliability. Generally, the po or

performances of

r W

(apart from

the

special

c a s e of

high

r ed un da nc y a nd

single B D ) make th e rN test wort h-con -

ceding

th e additional implementation effort.

Neverthe-

less,

th e la tt er

i s

no t reliable e n ou gh e it he r;

i n d e e d ,

in

case

of multiple

interacting

BD , th e one-to-one cor-

respondence

between

largest

I r N

and e rro ne ous m ea -

surement

stops

being

guaranteed

: valid


may

thus be declared

f als e a nd

vice-versa.

Note

that th e decision is

taken on

a

global

basis

given

by the sole detection

test,

which

just

informs

about

th e existence of BD among th e measurements, but

does

n ot i nd ic at e

whether

the

eliminated

o n e s are

actu-

ally

erroneous.

4.1.2. Assessment

P u o s

*

it

is sim ple , since the

only

computation

it

needs

be-

sides

estimation

i s

that of

residuals;

*

it i s capable to wa r n

th e operator

that

the

BD are

topologically

u n i d e n t i f i a b l e ,

provided

th e

method

of

§ 2 . 4

i s implemented.

*

it

i s

heavy since it requires a

series of reestima-

tion-detection a ft er e ac h

elimination;

this

m ay lead

to computer times incompatible

with

th e on-line

re -

quirements;

*

it

may

lead

to a

degradation of the measurement

c o n -

figuration

and a

subsequent

drop

of th e

power

of

th e

detection

test

( s e e

f i g . 1 ) ; t h i s in t ur n ma y

cause

an

important probability of

non-detecting remaining

BD

(especially w he n they be com e

critical);

*

it

can

provoke an undue elimination

of valid m e a s u r e -

ments causing not only

a

rough identification

but

also a

drop of th e detection test

power.

When

using

the rN test,

this

situation

arises in th e

case of

m ul ti pl e i n te r ac t in g

BD or of

BD located

in regions

with lo w local

r e d u n d a n c y , i . e .

i n the

case

o f s tr in-

gent

identification conditions.

On the other

hand,

th e

r w

test

ma y

lead to a degradation even

i n

mild

situations.

4 . 1 . 3 .

R em ar ks o n th e correction of measurements

Within t h e

p ro c ed ur e b y e li mi na ti on ,

two variants

may be distinguished.

The first

consists in

correcting,

after each

reestimation, th e

measurement

having

th e

largest

trNi[

by substracting

from

i t s

value th e

esti-

mate

e

- 1

e i

=

wi

r i

( 8 )

ii1

while

keeping

c on st an t t he

W

matrix.

The second v ar i an t c o ns i st s i n

correcting a group

of

s

selected measurements among th e

suspected

ones by

subtracting

from

their values

th e

estimates

where :

es

=5e

rs

= r r 5

( 9 )

s

:

denotes

th e selected measurements,

i s

t h e

corresponding

( s Xs )

-dimensional

sub-

matrix of W

r s

: i s

th e

corresponding

s-dimensional

subvector

of

r . .

To avoid

successive reestimations

of

th e

state

vector,the

following

correction

formula of

J ( ' )

pro-

posed

by Ma

Zhi-quiang

[ 1 2 ]

can

be

used

J(XC)=

J(x)

- s r s

es ' ( 1 0 )

H e r e

xc

i s

the new s ta te v ec to r

obtained

from

the mea-

surements corrected

b y

e s

( i . e . e l i m i n a t e d ) .

T h e r e f o r e ,

2

J

x c

has

a

x

_ d is t ri b ut i o n w i t h

( m - n - s )

degrees

of

freedom.



4/13

3 0 4 0

The advantages

an d

d ra wb ac ks o f th e

a bo ve te ch -

niques are

summarized hereafter.

P U 4 o

*

The

main

attractiveness of t hese

techniques

i s

that

th e

correction

does

not

affect

the

measurement

con-

figuration. Hence,

th e

gain

matrix can be

kept

con-

stant

during

th e

successive

reestimations of

the whole

ide nt ific a t ion pr oc e dur e , w hi le k ee pi ng

th e

goodness

of the minimization procedure convergence.

On th e con-

trary,

eliminating

BD

ma y

deteriorate

this convergence.

It ma y even happen

that

such a procedure which

con-

verges properly through th e above t e c h n i q u e , diverges

when

eliminating

th e

BD .

C o n s

I n addition

to the

weaknesses

of

thevery procedure

by

elimination

listed

a b o v e ,

t h es e c or r ec t io n

techniques

induce th e

following

disadvantages

As

fo r th e

single

correction-elimination

*

there is a risk

that

some measurements

previously

cor-

r ec te d b ec om e

erroneous

a ga in . I nd ee d, in

order

that

correction and elimination

to be

equivalent

at

each

step, al l

th e ( s - 1 )

previously c o r r e c t e d

measurements

must be c o rr ec t ed a ga i n

along

with the lastone

through

e q . ( 9 )

( s e e Ref. [ 2 1 ] ) ;

*

there is

a

greater risk to declare f a l s e a

valid

mea-

surement

because

of th e

approximation

of

the

normal-

ized residual.

I nd ee d, t he variances

of

th e residuals

computed

on th e basis

of

th e

i n i t i a l

W

matrix

are no

longer valid

s inc e the

residuals

of

the

non corrected

measurements are

equal

to those

resulting

from the

actual

elimination

of th e corrected measurements and

th e residuals of th e corrected ones are zero ( i f th e

correction

is

carried

out o nl y t h ro ug h

e q .

( 1 0 ) ) .

Concerning

the

grouped correction-elimination

*

th e

computation

time m ay

increase

significantly ( a n d

even

become

prohibitive with

the

n um be r o f

tim es the

linear system

g iven b y ( 8 ) and ( 9 ) i s solved), even

if

a

grouped residual search

i s

used.

4 . 2 .

IDENTIFICATION BY

N W C

4 .2 .1 . D es c ri pt i on

The NQ C

methodology

consists

in

minimizing

the

cost

function

m

J

( x )

=

f i

( r i / a i )

( 1

1 )

i = 1

where

f i

i s

equal

to

r l / 4 G

when

I

r X i


5/13

3 0 4 1

E [ 6 s i ]

=

esi

( 1 4 )

The

H TI

method

may

b e e xp lo it ed

through either

of

the t wo strategies

proposed

in

Ref.[21]

:

S t A a t e g y _ q

:

the decision is taken

witha

fixed

type

a

error

probability

of

declaring

false

a measurement

w h i c h

is valid.

S t A a t e g y _ _

th e

decision is

t a k e n with a f ix ed t ype

6

error

probability

of declaring valid

a measurement

w h i c h

is

false.

More

explicitly,this

strategy

consists

in

ad-

justing

the

parameter Vi

fo r each

selected

m e a s u r e m e n t

and

in

refining the successive

s

lists by

selecting

at

each cycle

only th e

measurements

which have

yielded

a

positive hypothesis

testing.

4.3.2. A s s e s s m e n t

Puz :

*

The

H T I

method is

generally

able to

identify

all

BD

within

a

si ng le s te p

( o r

at

w o rs t w it h in

two

steps).

This

is especially

true for

strategy

6

.

Concerning

strategy

a

,

experience

h as

s ho wn t ha t,

when all

th e

BD

have

not

b e en i d en t if i ed

by

the

first

test, a

sec-

ond one, performed

after

a

reestimation,

is

sufficient

to

complete

th e identification.

Note that

in both

strategies,

situations

where

all

BD have n ot b een

se-

lected

may

lead

to a

slightly larger

number

of r e e s t i -

mations.

*

This method

i s

able

to

identify strongly interacting

BD .

This important

advantage

results from

eq.(14)

which

shows

that,

unlike

the

residuals,

th e estimate

e s i

is

n o t

affected

by

th e

presence

of

BD

among

t h e

other

measurements.

In other

words,

the

very

n o t i o n

of interacting

BD

becomes

meaningless.

*

The

method

treats

properly topologically

unidentifi-

able

BD. Indeed,

the

procedure

of

§

2. 4

applies

to

the H IT method

as

well.

C o n z :

*

There

is a r isk

of

poor

identification,

corresponding

to

the

case w h e r e

one

or

several

BD are n o t s el ec te d.

This risk

can

however

be alleviated

through

appropri-

ate techniques

[ 2 1 ] .

*

T he m et ho d requires

the

computation

of

the

W s s

matrix,

whereas

the other procedures

merely

need

th e

diagonal

of

the

W

matrix.

Note

h ow ev er t ha t

t h e

technique

pro-

posed

in

[ 2 1 1

avoids necessity

of computing

the

com-

plete

E

matrix.

5 .

COMPARING

SIMULATION

RESULTS

5 . 1 .

SIMULATION

CONDITIONS

5 .1.1.

T h e test

systems

All the identification

methods

have

b ee n t es te d

on

two

test networks and

two

real

systems,

namely

the

IEEE

30-bus

an d

118-bus

n et wo rk s, a nd

a

B e lg i an 4 00 / 22 5 /

150/70

k V and the Tunisian 220/150/90

kV

power

systems.

F o r

the former

two,

the

m e a s u r e m e n t configurations

have

b ee n f ix ed randomly

a nd f ur th er adjusted

so

a s t o

comply

with observability

constraints

while keeping

an

overall

redundancy

of

about

2

. As for

th e

two others,

their

(actual)

configurations have

a

redundancy

of

1.9

( B e l -

gian)

and 2.8

(Tunisian).

The

variety

of

the

systems

characteristics (with respect

to size, topology,

elec-

trical

parameters and

measurement

locations)

allows

drawing

valid conclusions as

regarding BD

analysis.

F o r

purposes

of illustration,

the

well-known

IEEE

30-bus system is chosen

here;

its d ia gr am along with

the

adopted

measurement

configuration

and

characteris-

tics

are

shortly

described

in

the Appendix.

5 .1.2. The

tested

methods

T h e

results

reported

b e l o w

are

merely

concerned

with

the

most

important

v a r i a n t s o f

e a c h

of

th e

t h r e e

identification

methodologies.

S om e

specific

implementa-

tionquestions

are also

discussed.

I B E .

Because

o f

the

inappropriatness

o f th e

grouped

elimination,

only

t h e

single

e l i m i n a t i o n

scheme

is

con-

sidered

here.

However,

in o r d e r t o d e c r e a s e

th e

n u m b e r

of

successive

reestimations

(and hence

to save computer

time),

the

a ct iv e an d

reactive

m e a s u r e m e n t subsets

are

processed

in

parallel,

i.e. an a ct iv e a nd

a reactive

measurements

are eliminated a t the

same time

( a s

pro-

posed

in

Ref.

[ 7 ] ) . This

shortening

is based

on

the

hy-

pothesis

of

decoupling between

active

and reactive

vari-

ables

in E. H. V .

power systems.

NQC. When

th e detection

tests reveal

presence

of

BD

among

the measurements,

a

new estimation

is performed

based

on one

of

the

proposed

NQC.

T o overcome

th e diffi-

culty of

local minima, th e

s t ar ti n g p oi n t

of the itera-

tive

procedure

i s

th e

estimate

given by the WLS

estima-

tor ( a s proposed

i n Ref. [ 3 ] ) .

The

threshold

y -

which determines

the

transition

from

quadratic

to nonquadratic

estimation

has b e e n

taken equal

to

5 .

Experience h as sh own

that this

choice

i s

reasonable;

indeed a t o o

s ma ll v al ue

fo r this thres-

hold leads

to the

rejection

of

to o many measurements

a nd h en ce

to convergence

problems,

whereas a

to o large

v al ue r e su lt s in a

poor

BD

rejection.

The

study of NQC h as

n ot

b ee n e xt en de d

to the case

of

a threshold varying

during

th e iterative

process;our

experience

makes us think

that this

refinement

is no t

capable

of

significant improvements.

H T I . The

elements of

the

W ma trix n e e d e d

fo r th e com-

putation

of the

n o rm a li z e d r e si d ua ls

and for

the

Wss ,

submatrix

are

obtained

from the

available jacobian H

and gain

G

matrices.

In

practice,

H and G

are kept

constant

after th e first two iterations

i.e.

they

are

computed

and/or factorized

only

twice). Experience

has

shown th at th is does

no t

affect

the accuracy

of

W s s

provided

that

H

and

G are

kept

constant

at th e

same

iteration

step.

T he number

s

of selected


is arbi-

trarily limited

to 3 0 but -when

the

test

on J(Xc)

( s e e

§

4.1.3)

detects

the presence

of BD

among

the remaining

measurements, groups

of 1 0 additional


are

successively

appended

to

the

previous

selection.

Concerning

th e strategy

a

,

th e pa ra me te r

V has

been t ak en e qua l to

2

(a=

4.6 ).

The

choice of

a

h i g h e r

value

( 3 . 0

fo r example)

could

result

i n

an

incomplete

BD

identification;

indeed,

in the presen ceo f inaccurate

estimates

& S i

,

the corresponding

S error probability

is

to o

high.

This

is

one

of the

reasons

for

considering

strategy

6

.

As

fo r

strategy

$

,

the

parameters

of

concern

take

on the f o ll o wi n g v a lu e s

H e n

I e s . I =

40

,

5=

1

N=

-2.3 2 and

(N

Ia)max=

3

-

1

15

4 u

-

2 . j

v3

l I i i _ l

i

=

with

0<

vi<

3

I

( 1 5 ' )

5 . 1 . 3 .

The test cases

I n

o rd er for an

identification method

to

be prac-

tically effective,

it

has to

pass

the

e x a m on

multiple

BD .

The

cases

chosen

to be

reported

below pertain

to

th e three

possible

types

of

such

BD

1 s t case

:

multiple

interacting

BD

located

around

th e

same n o d e ;

2nd

case

: multiple

noninteracting

BD

having

very

dif-

ferent

m a g n i t u d e s

and

belonging

to

poor

an d rich

areas;

3 r d

case

:

topologically

unidentifiable

BD.

The above

list

is certainly no t

exhaustive

but nevertheless

suffi-

cient

to illustrate th e

considerations

of

Section

4 .

5 . 2 . F I R S T

C A S E

: M U L T I P L E

I N T E R A C T I N G

B A D D A T A

Four

interacting

BD

s u r r o u n d i n g

node

1

have

been

introduced.

Their

degree

o f i nt er ac ti on

is low

to

moder-

T A B L E

I C H A R A C T E R I S T I C S

O F

T H E

F O U R I N T E R A C T I N G

B D

B a d

d a t a

A c t u a l V a l u e

M e a s u r e d V a l u e

e 1 = z

- h j ( x )

e |

h i

( x )

z

e

F L P

1 - 2

1 7 7 . 3

0 . 0

- 1 7 7 . 3

1 1 8 . 2

F L Q

1 - 2 .

- 2 5 . 7

3 0 . 0 5 5 . 7

3 7 . 1

I N P

1

2 6 1 . 2 0 . 0 - 2 6 1 . 2

1 7 4 . 1

I N Q

1

- 2 7 . 1 3 0 . 0

5 7 . 1 3 8 . 1

,w



6/13

3 0 4 2

T A B L E I I

S U C C E S S I V E

L I S T S

O F S U S P E C T E D

M E A S U R E M E N T S

I N

T H E S IM PL E

E L I M I N A T I O N P R O C E D U R E T H R O U G H T H E r N T E S T

l s t e s t i m a t i o n 2 n d e s t i m a t i o n 3 r d e s t i m a t i o n 4 t h e s t i m a t i o n 5 t h e s t i m a t i o n

A c t i v e R e a c t i v e

A c t i v e

r N I

J R e a c t i v e rN| A c t i v e

r N i

R e a c t i v e

r N i

A c t i v e

r N .

R e a c t i v e

r N i

A c t i v e

r N

N - R e a c t i v e r N i

F L P 2 - 1 - 8 1 . 7 F L Q 2 - 1 2 8 . 0 I N P 2 - 7 1 . 8 I N Q

2

2 9 . 3 F L P 1 - 3 3 9 . 5

F L Q

4 - 3 1 0 . 8

I N P

1 - 1 0 . 6

I N I )

1 - 7 4 . 1

F L Q 1 - 2

2 2 . 7

F L P

1 - 3

5 6 . 6 F L Q 1 - 2 1 5 . 7 I N P 1 - 2 3 . 6 F L Q 1 - 3 - 6 . 3 F L P 4 - 3

1 0 . 6

I r N I l

<

3

I r N i l

<

3

I r N i l

<

3

F L P

1 - 3 4 9 . 8

IN Q 1 1 8 . 2 IN P 1

- 4 0 . 5 I N Q

5 1 3 . 4

F L P

4 - 3 - 2 2 . 1 F L Q

6 - 2 - 4 . 0 F L P 1 - 2

1 0 . 6

F l P

1 - 2 - 4 6 . 7

I N Q 2

1 7 . 0

F L P 4 -3

- 2 8 . 7

F L Q

1 - 3

- 1 2 . 0 F L P

1 - 2 1 0 . 5

F L Q

1 - 2

3 . 2

IN P

2

- 4 1 . 6 F L Q 1 - 3 - 1 0 . 5 F L P 1 - 2 - 2 4 . 0 F L Q 4 - 3 1 0 . 6 F L P 2 - 6 - 7 . 8

J()=

1 5 2 1 1 . 6 >

8 7 . 0

J ( 2 ? = 7 6 9 3 . 8

>

8 4 . 5 J ( ± )

=

1 7 5 3 . 3

>

8 2 . 1 J ( 2 ) = 1 5 6 . 2

>

7 9 . 6 J ( £ )

4 3 . 5

<

1 8 . 3

ate.

Their

characteristics are

given

in

Table I

( v a l u e s

i n M W / M V a r ) .

They

are

of

both

t y p e s ,

IN

( i n j e c t i o n )

and

FL

( f l o w ) ,

of

P / Q

(active/reactive p o w e r ) .

5 . 2 . 1 . Identification by elimination

5 . 2 . 1 . 1 .

E t i i n a t i o n

b o 6 e d o n

r N

The identification procedure

requires four succes-

sive

elimination-reestimation

c y c l e s ,

after th e alarm of

th e

detection

test. They

are

summarized in Table I I . The

elimination

of

th e

fourth active measurement

makes

crit-

ical

two others. T he final list

of m e a s u r e m e n t s

labelled

false

i s

thus

the

following.:

-

eliminated : FLP

2 - 1 ,

FLQ

2 - 1 ;

I NP

2 ,

INQ 2 ;

F LP

1 - 3 ,

FLQ

4 - 3 ;

INP

1 ;

-

become

critical

: F L P

4 - 3 ,

F LP 1 - 2 .

T he final state estimate i s

th e

one

obtained

at

th e

en d of

the

fifth

estimation;

some characteristic

values

ar e

reported

in

column four of Table

IV ( s e e next

page).

The

r es ul ts i ns pi re the following comments.

( i )

Both

erroneous

active measurements

ar e

present

in

th e final

l i s t ,

even if

one

of them has

been

included

thanks

to

th e

critical

measurement analysis.

( i i )

T hr ee v al id

measurements

have

i nc or re c tl y b ee n

de-

clared false.

( i i i ) None

of

the

two

erroneous

reactive m e a s u r e m e n t s

has

been

identified.

Indeed the

improper

elimination

of

three

valid

( r e a c t i v e )

d at a c au se d

an

important

weaken-

in g

of the measurement

configuration.

This

in

turn

pro-

voked a

decrease

in the

value

of

th e

W i i

coefficients

and

hence

in the

detection

capability,

as described in

§

2 . 2 .

A

more

d e ta il e d a na ly si s

of

this

question

i s

given

below.

( i v )

The final state

estimate

i s

completely

e r r o n e o u s

in a certain

neighbourhood

of node

1 ,

since

F LP

1 - 2 ,

FLQ

1 - 2

an d

IN Q

i

have not been eliminated.

I t i s

interesting

to

explore

f ur th er t he mechanism

of detection

capability

decrease

by considering

the de-

gree

of BD

interaction.

Let

e i ( r e s p .

e 2

)

be

the

weighted

error

affecting FLQ

1 - 2

( r e s p . INQ

1 ) .

We

de -

termine

th e domain

D 1

of

the

two-dimensional

space

( e j , e 2 )

in

which the

probability

to detect the

presence

of

BD

i s

smaller

than

a

given

value

Pd

( P d =

0. 9 here-

a ft er , h en ce

NPd

=1.28

) .

Using eq . ( 7 )

and

taking

into

a cc ou nt t ha t

Npd

=-NS

yields

I v < W l

e

e 2

<

X + N P

( 1 6 )

½

I 1 =

I21 e j

+

2

e

<

X + N P d

( 1 7 )

Substituting

into

( 1 6 )

and

( 1 7 )

the values

of the

W i j

coefficients

before

any

elimination

( s e e

Table

I I I )

yields

-

_ 4 . 2 8

<

0.886e;

-

0 . 3 1 0

e

<

4 . 2 8

( 1 8 )

-4.28

<

- 0 . 3 8 0 e j

+

0.724 e2

<

4.28

( 1 9 )

These

inequalities define the domain

D 1

plottedin Fig.3.

T A B L E

I I I

-

S U C C E S S I V E V A L U E S

O F

W - M A T R I X T E R M S

R E L A T I V E

T O B D

B e f o r e A f t e r e l i m . o f

A f t e r

e l i m i n a t i o n o f

a n y

e l i m i n a t i o n

FL Q

2 - 1

a n d

I N Q

2

FL Q 2 - 1 ,

I N Q

2

a nd

F L Q

4 - 3

F L Q

1 - 2

I N Q

I

F L Q

1 - 2

I N Q

1

F L Q

1 - 2

I , N Q

1

F L Q

1 - 2

0 . 7 8 5 0 . 5 0 5 6 4 - . 4 3 0

. 3 4 4

0 . 0 3 3 0

I N Q

1 -

0 . 2 7 5 0 . 5 2 4

- . 4 3 0 0 . 3 8 2

-

4 . 3 3 0

0 . 3 3 6

P

>90

The relatively restricted extent of

D ,

denotes

a

good

ability

of

BD

detection.

On the

other

hand, substituting

th e

values

of

t h e

W i j

coefficients

after

elimination

of

F LQ 2 - 1 ,

INQ

2

and

FLQ

4- 3 ( s e e Table I I I )

gives

-4.28

<

0 . 5 8 7 e l

-

0.563 e2

<

4.28

-4.28

<

- 0 . 5 6 9 e i

+

0.580 e2

<

4.28

( 2 0 )

( 2 1 )

The corresponding

domain

D 2

i s

plotted

in

Fig.4.

On e

can

see that D2 i s

notably larger

than

D l

.

This

illus-

trates

th e

drop

of

th e

detection power test.

Note

that

th e

actual value

of

the

two

BD

( s e e

Table

I )

are

locat-

ed

j u s t

i n

D 2 ; this explains why

they

are

n o

l o n g e r

detected. Table

II I

shows

th e successive decrease

in

th e terms

of

concern

of

W matrix

resulting

from th e

successive eliminations,

and

hence

the

corresponding

increase

in the

degree

of

BD interaction.

5 . 2 . 1 . 2 .

E U i n a L t i o n

b o 6 e d

o n

r w

The results and

th e

conclusions

are

similar

except

that measurements

are not eliminated

in

th e

same order:

-

e l i m i n a t e d

:

F L P

2 - 1 , F L Q 2 - 1 ; F L P

1 - 3 , I N Q 2 ;

I N P

2 ;

F LP

1 - 2

F LQ 1 - 3 ;

-

become

critical :

I NP

1 ,

FLP 4 - 3 .

Moreover,

the

corresponding

domains D 1 and D 2

are

larger

than

in the-previous case.

5 . 2 . 2 .

I de n ti f ic a ti o n b y NQC

The

state estimates

g iven b y

th e

Q T ,

QL

and

QR

criteria through

the residuals

rW

ar e

reported

in

Table

IV

along

with

th e

a c t u a l

values of

th e

corresponding para-

meters.

Table

V lists

the suspected

measurements

( i . e .

those

characterized

by

I r W i J >

3 )

obtained

after

estima-

t i o n . The

s al ie nt r e su lt s

are

th e following.



7/13

3 0 4 3

T A B L E

I V

E S T I M A T I O N R E S U L T S

P R O V I D E D B Y

N Q C

A N D

B Y I B E

M ET H O D S ( M W , M V a r ,

p . u . ,

d e g r e e )

E l e c tr i c a l A c t ua l

I B E

N Q C

v a r i a b l e s

v a l u e s _ - - - - - - - - - - - - - T - Q - r -

Q

____

r w r N

Q T

Q L

Q R

M O D

1

1 . 0 6 0

1 . 0 5 2 1 . 0 5 8

1 . 0 6 5

1 . 0 6 3 1 . 0 5 8

F L P

1 - 2 1 7 7 . 3

- 1 9 . 9

0 . 0

1 5 5 . 9

1 6 6 . 5 1 7 4 . 6

F L Q 1 - 2

- 2 5 . 7

2 6 . 6

3 1 . 4

- 6 . 2 - 1 0 . 4

- 2 0 . 7

F L P

1 - 3 8 3 . 9 2 0 . 0

2 8 . 3 7 1 . 2

7 7 . 4

8 2 . 6

F L Q

1 - 3

- 1 . 4

6 . 3 - 2 . 7

7 . 6

4 . 9 0 . 9

I N P

1

2 6 1 . 2

0 . 0 2 8 . 3 2 2 7 . 1

2 4 3 . 9

2 5 7 . 1

I N Q

1

- 2 7 . 1 3 2 . 9 2 8 . 7

1 . 4

- 5 . 5 - 1 9 . 8

M O D 2 1 . 0 4 5

1 . 0 4 0

1 . 0 4 0 1 . 0 4 2

1 . 0 4 1

1 . 0 4 0

P H A

2 - 5 . 5

0 . 9

0 . 3

- 4 . 7

- 5 . 0

- 5 . 4

I N P

2 1 8 . 3 2 1 0 . 2

1 9 0 . 2 2 5 . 8 2 2 . 4

2 0 . 4

I N Q

2

3 1 . 9

- 3 6 . 6 - 4 0 . 2 2 0 . 2

2 0 . 6 2 9 . 7

M O D

3

1 . 0 3 3

1 . 0 2 9

1 . 0 4 8 1 . 0 2 5 1 . 0 2 6 1 . 0 2 7

P H A

3

- 8 . 1

- 1 . 7 5

- 2 . 7

- 6 . 7 - 7 . 4 - 8 . 0

I N P

3 - 2 . 4 5 9 . 1

5 1 . 1

9 . 7

3 . 8

- 1 . 4

i N Q 3

- 1 . 2 - 1 8 . 4

4 7 . 3 - 1 2 . 6

- 8 . 4 - 2 . 5

M O D

4

1 . 0 2 7

1 . 0 2 2

1 . 0 2 1

1 . 0 1 8

1 . 0 1 9

1 . 0 2 0

P H A

4 - 9 . 8

- 3 . 4 - 4 . 0

- 8 . 4 - 9 . 1 - 9 . 7

I N P

4 - 7 . 6 - 6 . 3 - 6 . 2

- 1 . 3 - 3 . 8

- 5 . 8

I N Q

4

- 1 . 6

- 4 . 5 - 6 1 . 2 - 1 1 . 2 - 8 . 1

- 6 . 5

T A B L E

V

S U S P E C T E D


B Y

N Q C

A L O N G

W I T H T H E I R

r W i

O B T A I NE D A F T ER

E S T I M A T I O N

N Q C

S u s p .

m e a s u r t s .

r W l

S u s p .

m e a s u r t s .

r W _

I N P

1

- 1 5 1 . 4

F L Q

1 - 2

2 2 . 1

F L P

1 - 2 - 1 0 3 . 9

I N Q

1

1 9 . 1

F L P

2 - 1 - 1 1 . 8 F L Q

2 - 1 1 3 . 8

Q T

F L P

1 - 3

1 0 . 0 I N Q 2

7 . 9

I N P

2 - 4 . 4 F L Q 1 - 3 - 7 . 7

F L P

6 - 2

- 3 . 2

F L Q

4 - 2

3 . 7

F L P

2 - 5 3 . 1

I N P 1

- 1 6 2 . 6

F L Q

1 - 2

2 6 . 9

F L P

1 - 2

- 1 1 1 . 0 I N Q 1

2 3 . 7

Q L

F L P 1 - 3 5 . 9

F L Q 2 - 1

9 . 8

F L P

2 - 1

- 5 . 2 I N Q

2 7 . 7

F L Q

1 - 3

- 5 . 9

I N P

1 - 1 7 1 . 4 F L Q 1 - 2

3 3 . 8

O R

F L P

1 - 2

- 1 1 6 . 4

I N Q 1

3 3 . 2

F L Q 1 - 3

- 3 . 3

( i )

The

BD

have

not been

completely rejected

and th e

final

state estimate

i s still erroneous

in th e

vicinity

of

node

1

( s e e

Table

I V ) .

( i i )

Therefore,too many

valid measurements

are

suspect-

ed

at

the end of

th e

estimation. Note

that

th e

stronger

the

rejection ( a s

fo r

example

for the

QR

c r i t e r i o n ) ,

the smaller

the list of

suspected

measurements

( s e e

Table

V ) .

( i i i )

Except

f o r th e

QC

criterion

which has

shown un-

able

to

provide

an

estimation,

al l the

other

NQC

have

required

a

great

-

i f not

prohibitive

-

number of itera-

tions

( s e e

T ab le X b e l o w ) .

T hi s s low c on ve rg en ce

i s du e

to

the

r e j e c t i o n

of

al l measurements

around

nodel

which

in turn tends

to make th e network

numerically

unobserv-

able.

The

QC

criterion is

particularly

unreliable since

by

eliminating

al l th e

suspected

measurements

i t

makes

th e

network

topologically

unobservable.

( i v )

All th e

NQC diverge

if th e

gain

matrix i s

kept

constant

after

th e

first

two

iterations.

T h u s ,

unlike

for the WL S

estimation,

t hi s m at ri x

has been

computed

at

each

c y c l e .

5 . 2 . 3 .

Identification by

H TI

Among

th e 3 1

suspected

measurements

given

by

the

r N

test,

only

2 5

are

chosen

( s =

2 5 ) .

Indeed th e

6

re-

maining

ones

( I N P

2 ,

FLP

6 - 7 ,

INP

5 ,

F LP

4 - 3 , FLQ 4 - 3 ,

FLP

4 - 1 2 )

are

necessary

to ensure th e

observability

of

the

system ( i . e .

t h e y

would

become c ri ti ca l a ft er

elim-

inating

th e

2 5

above-mentioned

m e a s u r e m e n t s ) .

Computa-

T A B L E

V I

F I R S T

S E L E C T I O N R E S U L T S O F H T I

T H R O U G H

S T R A T E G I E S a

A N D

, B .

N U M B E R O F

S E L E C T E D


2 5

1 s t S e l e c t i o n S t r .

a

S t r a t e g y

8

S e l e c t e d r .

v

x

m e a s u r e m n e n t

e s i

e s i

i

r 1 1 1 i

x i

F L P 2 - 1

2 . 3 0 - 2 3 . 2 7

9 7 . 4 7 1 0 5 6 . 0 0

0 . 0 0 0 . 0 0

I N P

1 - 2 6 1 . 2 0

- 2 1 1 . 6 5 1 7 3 . 5 1

3 3 4 5 . 0 0 0 . 0 0

O . 0 0

F L P

1 - 3 2 . 3 3

2 3 . 5 1

7 3 . 8 6

6 0 6 . 2 0

0 . 0 0 0 . 0 0

F L P

1 - 2

- 1 7 7 . 3 0 - 1 4 8 . 8 9 9 9 . 7 5 1 1 0 6 . 0 0 0 . 0 0 0 . 0 0

F L Q

2 - 1

- 2 . 8 6

1 9 . 3 4 4 4 . 7 5

2 2 2 . 5 0 0 . 3 7

8 . 2 8

F L Q 1 - 2 5 5 . 6 9

4 1 . 8 4 3 9 . 3 9

1 7 2 . 4 0 0 . 7 3 1 4 . 3 8

F L P

4 - 2 1 . 0 9 - 9 . 6 7

4 1 . 9 4 1 9 5 . 4 0

0 . 5 5

1 1 . 5 3

F L P

6 - 2 - 0 . 6 4 - 8 . 7 9 3 4 . 3 7

1 3 1 . 2 0

1 . 1 8 2 0 . 2 8

I N Q 1

5 7 . 0 6 3 9 . 7 4

5 3 . 5 1 3 1 8 . 1 0

0 . 0 0 0 . 0 0

F L P

2 - 6 - 1 . 8 3 7 . 0 3

3 5 . 6 4

1 4 1 . 1 0 1 . 0 6

1 T . 8 9

I N Q

2

- 0 . 7 8 1 7 . 6 3 4 2 . 9 9 2 0 5 . 3 0

0 . 4 8 1 0 . 3 2

F L P

2 - 5 1 . 6 1 5 . 9 2 1 9 . 7 0

4 3 . 1 2 3 . 0 0 2 9 . 5 5

F L Q 1 - 3 - 2 . 6 7

- 6 . 1 5 1 6 . 1 2 2 8 . 8 9 3 . 0 0

2 4 . 1 9

F L Q

4 - 2 0 . 2 3 4 . 9 5

1 5 . 1 5 2 5 . 5 0

3 . 0 0 2 2 . 7 2

F L Q 6 - 2 - 1 . 5 9

2 . 9 4 1 5 . 5 4

2 6 . 8 3 3 . 0 0

2 3 . 3 1

F L P 6 - 8

1 . 2 9 1 . 3 7 1 1 . 7 6 1 5 . 3 7

3 . 0 0 1 7 . 6 4

F L Q

6 - 7

0 . 7 0

- 6 . 7 2 2 1 . 0 6

4 9 . 3 0 3 . 0 0

3 1 . 6 0

I N Q

5 1 . 8 2

- 6 . 3 1 2 2 . 3 6

5 5 . 5 5 3 . 0 0

3 3 . 5 4

F L Q

2 - 6 0 . 3 0 - 2 . 1 4

1 2 . 3 2 1 6 . 8 8

3 . 0 0 1 8 . 4 9

F L P

4 - 6 - 0 . 3 0 - 1 0 . 1 2

2 9 . 0 1 9 3 . 5 2

1 . 8 3 2 6 . 5 5

F L Q 6 - 8

- 0 . 7 9

- 0 . 1 8 1 1 . 7 1 1 5 . 2 4

3 . 0 0 1 7 . 5 7

F L P 6 - 4

- 0 . 4 2

9 . 2 2 2 8 . 5 0

9 0 . 2 3 1 . 8 3

2 6 . 0 7

F L Q

2 - 5

- 0 . 0 1

0 . 3 8 8 . 3 2

7 . 6 8 3 . 0 0

1 2 . 4 8

F L Q

4 - 6

1 . 3 1

1 . 2 5

4 . 5 6 2 . 3 1

3 . 0 0 6 . 8 4

F L P 6 - 9 1 . 2 2

1 . 0 2

4 . 3 3 2 . 0 8 3 . 0 0 6 . 4 9

T A B L E V I I

S t r a t e g y a : 2 n d

S e l e c t i o n

S e l e c t e d m e a s u r e m e n t s

e S i

X

F L P 2 - 1 4 . 4 7

4 . 9 1

I N P

1 - 2 6 1 . 2 3

6 . 6 4

F L P

1 - 2 - 1 7 7 . 2 8 4 . 9 8

F L Q 2 - 1 4 . 9 1

2 0 . 7 2

F L Q 1 - 2

5 4 . 3 2 2 0 . 6 8

IN Q

1

5 7 . 4 2

2 6 . 6 3

I N Q 2 7 . 8 6 2 3 . 3 0

F L Q

1 - 3

- 0 . 9 4

7 . 0 4

F L Q

4 - 2 1 . 3 7 3 . 8 4

F L Q 6 - 7 - 3 . 5 7

5 . 4 6

- F L Q

2 - 6

0 . 3 2

3 . 7 8

T A B L E

V I I I

S t r a t e g y

B:

2 n d

s e l e c t i o n S t r a t e g y

8 : 3 r d

s e l e c t i o n

S e l e c t .

e s r j vi

X i

S e l e c t .

d s

|

r i j

v i

X i

M e a s .

M

i~i~

e a s .

j

1 1

1

F L P 2 - 1

0 . 8 0 4 . 3 4 3 . 0 0

9 . 3 8

I N P 1

- 2 5 9 . 0 4

3 . 0 3

3 . 0 0

7 . 8 3

IN P

1 - 2 5 4 . 7 3

9 . 1 9

3 . 0 0

1 3 . 6 4

F L P 1 - 2

- 1 7 2 . 7 9 2 . 0 4 3 . 0 0

6 . 4 3

F L P 1 - 3 5 . 0 3

2 . 1 7

3 . 0 0

6 . 6 3

IN P

2

- 0 . 8 3 4 . 0 1 3 . 0 0 9 . 0 1

F L P 1 - 2 - 1 7 3 . 5 0

4 . 5 0

3 . 0 0 9 . 5 5 F L Q

1 - 2

6 0 . 5 1 1 . 6 0 3 . 0 0

5 . 6 9

F L Q 2 - 1 1 0 . 1 8 4 7 . 4 5

3 . 0 0 3 1 . 0 0 I N Q 1

6 4 . 5 1

2 . 4 5 3 . 0 0

7 . 0 4

F L Q

1 - 2

4 9 . 3 1 4 6 . 9 9

3 . 0 0 3 0. 8 5 F L P 4 - 3

- 2 2 . 3 9 3 0 . 0 8 3 . 0 0

2 2

I N Q

1

5 3 . 3 5 4 7 . 9 9 3 . 0 0

3 1 . 1 7

I N Q

2 1 4 . 7 2

5 2 . 9 7

3 . 0 0 3 2. 7 5 S t r a t e g y

8 :

4 t h s e l e c t i o n

F L P 6 - 7 3 . 7 6 4 . 6 8

3 . 0 0 9 . 7 4

1

IN P

5

7 . 0 7 9 . 6 3 3 . 0 0

1 3 . 9 6

I N P

1

- 2 5 7 . 2 5

2 . 5 4 3 . 0 0 k

7 . 1 8

F L Q

4 - 3

8 . 5 1 1 2 1 . 0 0

1 . 3 3 2 1 . 9 4 F L P

1 - 2

- 1 7 4 . 5 1

1 . 6 5

3 . 0 0

I 5 . 7 9

F L P

4 - 1 2

2 . 0 0

2 . 2 4 3 . 0 0

6 . 7 4 F L Q 1 - 2 6 0 . 3 4

1 . 6 0

3 . 0 0 5 .

6 9

IN Q 1 6 4 . 9 1

2 . 4 4 3 . 0 0

7 . 0 3

tion

of

J ( ; c )

relative

to th e corresponding

( m - s )

m e a -

surements

gives

J G u c )

=

15211.6-

15178.3

=

3 3 . 3

J ( : i E c )

is

c h i - s q u a r e d

with

( m - n ) - s

=

118-59-25= 3 4 de-

grees

of

freedom.

The

threshold

corresponding

to

a risk

a=

1 %

i s

55.3

Hence

the test

on

J ( 2 c )

i s

n e g a t i v e

on e concludes

( w i t h

o f c our se

a

certain

error probabi-

lity )

that

there are no

m ore BD

among

th e

remaining

redundant measurements

( b u t

not

necessarily

among

the

six

above-mentioned o n e s ) .

T he r es ul ts

corresponding

to strategy

C a are

re-

ported

in

Tables VI

and

VII.

As can

be

seen,

only

three

BD have

been i de nt if ie d b y

th e

first

test.

The

fourth

one ( I N Q

1 )

has

not,

because

of

a too

h i g h

e r r o r

p r o b a -

bility

(ii

=

3 1 8 . 1

,

hence

=

45

)

.

These

three

measurements

are eliminated

and

the state

i s

estimated

ag ain. T he

s e co n d s e le c ti on i s

co mpo sed of ei gh t

new

s u s p e c t e d

measurements

a l o n g

with

th e

three

p r e v i o u s l y

eliminated ones .

The

identification

i s now

c o r r e c t l y

p e r f o r m e d .



8/13

3044

T A B L E I X

C H A R A C T E R I S T I C S

O F T HE E I G H T

N O N I N T E R A C T I N G B D

A c t u a l M e a s u r e d

v a l u e

v a l u e

e i

=

z i - h i ( x )

l e s i l W i i

h i ( x )

Z i

F L P

2 - 5

8 2 . 6 1 8 4 . 6 1 0 2 . 0 6 8 . 0

0 . 8 4

F L Q 2 - 5 2 . 8

1 0 1 . 7

9 8 . 9

6 5 . 9 0 . 8 5

F L P

1 2 - 1 5 1 7 . 6

6 9 . 2

5 1 . 6 6 4 . 5

0 . 1 5

F L Q

1 2 - 1 5

7 . 0

5 6 . 1

4 9 . 1 6 1 . 4

0 . 1 6

F L P 2 4 - 2 5 - 0 . 5 1 9 . 0 1 9 . 5 2 4 . 4

0 . 6 2

F L Q 2 4- 25

2 . 5

2 2 . 4

1 9 . 9

2 4 . 9

0 . 6 4

I N P

2 9

- 2 . 4

- 1 2 . 1

- 9 . 7

1 2 . 1

0 . 4 7

I N Q

2 9

- 0 . 9

- 1 0 . 2 - 9 . 3

1 1 . 6 0 . 4 7

Tables V I

an d VIII

summarize th e

results

of

strat-

egy 6

.

Four

cycles

of

selection

were

n ee de d. T he

si x

suspected

measurements

which w e r e

not

inserted

in

th e

f ir st s e le ct i on

( f o r

observability

r e a s o n s )

are intro-

duced

i n

the

se co nd a nd third

o n es.

Note that

fo r

th e

first

test,

th e value of

' i

i s equal to

zero

for five

measurements

:

this r es ul ts f ro m

th e

poor

accuracy

of

th e

correspondinq

estimates.

However,

fo r most of th e

measurements,

V i

reaches it s m ax im al v al ue

( 3 . 0 ) atthe

second

test.

This shows th e

rapid

increase

in-accuracy

of th e estimates

a nd h en ce

in

power

of

th e

identifica-

tion test.

Finally,

th e f ou rt h s el ec ti on i s

simply

com-

posed of th e four BD.

5 . 3 . SECOND

CASE

:

MULTIPLE

NONINTERACTING

BA D DATA

E i gh t n o ni n te r ac t in g

BD

h av e b ee n s im ul at ed .

Table

IX

lists

their

characteristics along with th e

values

of

th e diagonal

terms

of

W

matrix, w hi ch i nf or m about th e

quality of th e

corresponding

local

redundancy ( p o o r

fo r

1 2 - 1 5 ,

moderate fo r

2 9 ,

moderate

to

high

fo r the

o t h e r s ) .

5 . 3 . 1 . Identification b y e l im in a ti on

5 . 3 . 1 . 1 .

I B E

b a z e d

o n r N

The procedure has required

5

successive

c yc le s c or -

responding

to the

following final list :

-

eliminated

:

INP

5 , FLQ 2 - 5 ;

F LP

2 - 5 , FLQ

1 2 - 1 5 ;

F LP

1 2 - 1 5 ,

FLQ 2 4 - 2 5 ;

FLP 2 4 - 2 5 , IN Q

2 9 ; INP

2 9 ;

-

become

critical :

INP

2 .

All

th e BD have b e en e li mi na te d.

The incorrect elimina-

tion of

INP

5

has made

INP

2

critical. Note

that

the

latter measurement i s not

e r ro ne ou s; h o we v er this cannot

be

verified a posteriori.

5 . 3 . 1 . 2 . I B E

b a e d

o n

r w

The

identification has

required

7

s uc ce ss iv e r e es -

timations.

The f in al list

o f m e as ur e me n ts declaredfalse

i s th e

following.:

-

eliminated

:

F LP & F LQ

2 - 5 ;

FLP&FLQ

2 4 - 2 5 ;

F LP

1 2 - 1 4 ,

FLQ 1 2 - 1 5 ;

FLP

1 2 - 1 6 , INQ 2 9 ;

FLP&FLQ

4 - 1 2 ;

F L P 1 0 - 1 7 ;

MODV

1 3 ;

IN P

2 9 ,

MODV

1 2 ;

-

become

critical

: FLP

1 2 - 1 5 ,

INP

1 6 ,

INP 1 7 .

Seven valid

measurements

have improperlybeen

eliminated.

These undue eliminations

are

essentially caused

by FLP

&

FLQ 12-15,

which

are

located in a region of

lo w lo ca l

redundancy

( W i i =

0.15).

Moreover,

the final

estimate

is

erroneous

since

one

BD has

not been rejected;

indeed,

the l at te r h as

become

critical ( i t

has

been

labelled

false

as

i s

explained in § 2.4).

5 . 3. 2. I d en t if i ca ti o n

by NQC

Conclusions ar e similar

to

those drawn

for

th e pre-

ceding

case, even if

th e

ide ntific a tion c ondit ions

are

l es s s tr in ge nt here.

As in th e

interacting

case, the

QC

has been

unable to provide an

estimation.

Note

that,

because of a lo w

local redundancy,

th e

quality of

th e

state estimation in

th e

vicinity of node

1 5 i s

rather

bad for

al l

the NQC

( s e e Table X ) .

It

i s

worth-mentioning that the NQC

efficiency

i s

found to

vary

with

th e

noise

attached

to th e

valid measurements. This gives NQ C

a capricious

behaviour.

5.3.3. Identification by H TI

Both

s tr at eg ie s h av e i de nt if ie d

in one step

all the

8

B D .

This

identification has required a

s ing le t es t for

strategy

o t ,

and

4

successive

cycles fo r

strategy

a

.

5 . 4 .

T H I R D

CASE

: TO PO LOG IC A L L Y U N IDE N T I F IA BL E

BA D DA TA

A

g ro ss e rr or

has

been

introduced i n

th e

value of

F LP

1 0 - 2 0 .

This measurement

is

redundant

only

with F LP

1 9 - 2 0 .

The elimination method has drawn

up a

list

com-

prising

both

measurements.

The

H TI method has led to th e

same conclusion.

On the

contrary some

NQC

tendto

reject

F LP 19-20

and

to

keep

F LP 10-20.

5 . 5 . SUMMING

U P

S IM U LA T ION R E S U LT S

Table X summarizes

the

salient

simulation results

of

this Section,

along

with

computer

times

given

here

f o r

information only. I ndeed, m an y pa ra me te rs

- and

es-

pecially system's

size

-

influence

significantly

the

speed

of the various

identification

methods.

F or e x a m -

p l e ,

in

th e cases considered

here th e

reduced

s y s t e m ' s

size

is to

th e

advantage

of

th e

IBE

methods since

gener-

ally

t he y r eq ui re

many

state

reestimations.

Note

that

fo r

the

IB E

method based

on

rN ,

th e

Sherman-Morison formula

and

th e

sparse

inverse

matrix

method

proposed

in

[ 4 , 2 2 ]

have

been

used.

Note

also that

th e number

of th e

Z

matrix

' t e r m s

necessary

to be com-

puted fo r t h e

ET I

method

has been

assessed with

respect

to

1 7

and 49

state variables

respectively

for

th e inter-

acting

and

noninteracting

BD cases.

The

latter

should

be

regarded

as an

upper

bound.

The

simulations

h ave b ee n

performed

on

a

DEC 2 0

computer.

T A B L E

X S A L I E N T

S I M U L A T I O N

R ES U L T S O F

T H E

V A R I O U S

I D E N T I F I C A T I O N

M E T H O D S

4

I n t e r a c t i n g B D

8

N o n i n t e r a c t i n g

B D

M E T H O D

PERFORMANCE

I B E N Q C H T I

I B E N Q C

H T I

C R I T E R I A

\

W

r N

Q T

Q L

Q R

a

8

r w

r N

Q T

Q L

Q R

a

8

M e a s u r e m e n t s

A c t u a l B D

2

2

4

4

4

4

4

8

8

8 8

8 8

8

l a b e l l e d

f a l s e V a l i d d a t a

7 7

9 5

1 0

0

9

2

1 5 2

2

0

0

Q u a l i t y t o f

b a d

b a d b a d

r a t h e r

f a i r l y

g u d

g

b

go

a d

r a t h e r r a t h e r

g o o d

g o o d

s t a t e

e s t i m a t i o n

b a

a

a

a

o d g b a d

o o d

a godbd

b a d b a d

god

od

[

[ T ;

N u m b e r

o f

| 4

4

1

|

2

1

7 5

1 1

1

1

1

s t a t e

r e e s t i m a t i o n s

E

N u m b e r o f

___ __ _ __

.__ _____

_

_ _ __

u b e r o f

2

2 2 3 2 4

5 3 3

3

3 8 8

1 0 3 3

c r

i t e r a t i o n s / e s t i m a t i o n

m

T i m e

i n

s e c .

C P U

1 . 8

2 . 5

5 . 0

5 . 5

1 1 1 . 9

1 . 4

3 . 2 3 . 1

1 . 7 1 . 7 2 . 2 1 . 5

1 . 7

N u m b e r

o f

Q

t h e

Z

m a t r i x t e r m s

4 5 5

-

5 6 0 5 6 0

4 5 5

1 3 6 0 1 3 6 0

t o b e

c o m p u t e d

:___.



9/13

3 0 4 5

6 . CONCLUSION

The identification techniques

available t od ay h av e

been

classified i nto t hr ee

broad c la ss es ; t he ir capabi-

lityto face various type s of BD has

b ee n f oun d to differ

significantly

from

one

class

to another.

The NQC

exhibit

th e most

poor performances;

they

are

very

sensitive to

lo w

local

redundancy

and

to

inter-

action of BD;

they

have a slow

convergence

and

a

vi -

c i o u s . b e h a v i o u r .

In

brief, they

don't

show

to

be suit-

able

enough.

On th e other

h a n d ,

th e

IB E

techniques are attrac-

tive

with respect to implementation

considerations:

they are easy to

use a nd s im pl e to

implement.

T he y show

to

be

quite interesting as

long

as

the BD are non-

( o r

w e a k l y )

interacting

and located

in regions

of

moderate

redundancies. They

start

being

unefficient, however,

when th e

number

of BD and their spreading increase and

when

th e local

redundancy

decreases. Although

much more

reliable

than

th e NQC, th e IBE

m et ho ds l ea d to

inaccu-

rate BD

identification results at a c e rt a in l ev el

of

severity of

th e

identifiability conditions.

The

HTI

method, finally,

seems to combine effec-

tiveness,

reliability

and

compatibility with

on-line

implementation requirements.

T h is l at te r a s pe c t r e ce i ve s

at

p r e s e n t

further

consideration.

REFERENCES

[ 1 ] F.C.

Schweppe, J . Wildes,

D.B.

Rom,

Power

System

Static

State

Estimation. Parts

I ,

I I , I I I ,

IEEE

Trans.

on

PAS, v o l . P A S - 8 9 , N o . 1 , J a n . 1 9 7 0 , p p . 1 2 0 - 1 3 5 .

[ 2 1 J.F.

Dopazo,

O.A.

Klitin, A.M. Sasson, S t a t e Esti-

mation

for

Power Systems :

Detection and

Identifica-

tion

of

G r o s s Measurement

E r r o r s ,

Proc. of

th e

8th

PICA

C o n f . , Minneapolis, 1 9 7 3 , pp .

3 1 3 - 3 1 8 .

[ 3 ] E . Handschin, F.C. Schweppe, J .

Kohlas,

A.

Fiechter,

Ba d

Data

Analysis

fo r Power

System

State

Estima-

t i o n ,

IEEE

Trans. on

PAS, v o l . P A S - 9 4 , N o . 2 ,

M a r c h /

April

1 9 7 5 ,

pp.

3 2 9 - 3 3 7 .

[ 4 1 A . Merlin,

F.

Broussole,

F a s t

Method

fo r Ba d

Data

Identification in

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