External Equivalent

EE 521 Analysis of Power Systems

Chen-Ching Liu, Boeing Distinguished ProfessorWashington State University

EXTERNAL EQUIVALENT

Each power system (area) is part of an interconnectedsystem. Monitoring devices are installed and data aresystem. Monitoring devices are installed and data are available from ones own system. However, security analysis depends on accurate and reliable load flowanalysis depends on accurate and reliable load flow models of the non-monitored parts of the system as well.Approximate reduced models for load flow studies of theApproximate reduced models for load flow studies of the non-monitored parts are called External Equivalents.

InternalSystem ExternalSystem

(normal loadflow

System(equivalent

d l)model) model)

Boundary Buses

External2423 26

2529 30

e18 19

2022

27

2815

bBoundary

14

20

1617

21

28

b

i10

14

iInternal13 12

86

911

4IEEE 30-Bus System

3 7

1 2 5

1. Ward Equivalent (=1948)i : internal i : internalb : boundary

t l

e : externalBy nodal analysis. (YE = I)

b ie

e Y bY 0 E I (1)

=

e

b

eeY ebY 0

beY biYibb

ebb YY

eE

bE

eI

bI (2)

i 0 ibY iiY iE iI (3)

By (1):ebebeee IEYEY

eeebebeee

ebe

IYEYYE11

By (2):ie

IEYEYYEY )(

eeebebebeebe

bibibbbbbebe

IYYEYYY

IEYEYYEY11

)(

iebibib

ibb

ebb

eb

IEYEYY1

)(

eebeb

ibibebeebeibb

ebb

IYYI

EYEYYYYY1

1][

ebeebeebbeq

eeebeb

YYYYY

IYYI

1

1 :Let

eeebeeq IYYI1

=eq

ibb YY biY bE eqb II

(4)=ibY iiY iE iI

(4)

In block diagram:

Internal System (i)

WardEquivalenty ( )

ibbY biY

eqYibY iiY

eqY

Four Basic Steps in the construction of a wardE i l tEquivalent:1. Determination of the external network data

from available information2. Obtaining the ward equivalent network eqYg q

by Gaussian elimination.

3 Using the values for the complex voltages3. Using the values for the complex voltages at the boundary buses from the internal state-estimation to compute the flows in thestate estimation to compute the flows in the Ward equivalent branches.

4. Boundary matching, i.e., adding fictitious i j ti t th b d b th tinjections at the boundary buses so that Eq.(4) holds.

Base case: results of the state estimation (load (flow) for the internal system without contingency.g y

Why are the original and reduced modelsWhy are the original and reduced models equivalent?

In the base case, if one solves theload flow (4), the solution [ ] (systemstate) is exactly the same as that of Eqs. (1)-(3).

ib EE ,

) y q ( ) ( )The external equivalent model can be used in thecontingency evaluationcontingency evaluation.

Ward Equivalent gives reasonably accurate q g yresults for real power flows, whereas the accuracy for reactive power flow is relatively y p ypoor.

(This is due to the fact that the change in reactive power injection to maintain constantreactive power injection to maintain constant voltage at external PV buses is not accounted for )for.)

2. Ward-PV Equivalent

The Ward reduction process is applied only to external PQ busesexternal PQ buses.

The external PV buses are retained. The Ward-PV equivalents give excellent results

for contingency evaluation.

Given the system block diagram:Given the system block diagram:

I l QInternalsystem

Q

~

Boundary Externalsystem

Q Q Q V b ie

Q Q Q

Q

V b

QQY QVY QbY 0e

V

b

VQY VVY VbY 0bQY bVY

ieYY biY

b

ibQY bVY bbbb YY biY

0 0 ibY iiY

(Node (Bus) Admittance Matrix Y)

To obtain the ward-PV equivalent, one needs to do Gaussian elimination to eliminate Q part, p ,i.e.,

Q V b i

Q

V wvvvYwvvbY 0

(+)b

i

0 wvbvY

ibb

wvbb YY biY

(+)

Th W d PV i l t t k

i 0 ibY iiYThe Ward-PV equivalent network:

Internal ~VB

system ~

3. Extended Ward

To combine the simplicity of the Ward equivalent with the response of the Ward PVequivalent with the response of the Ward-PV equivalent.It i W d i l t ith dditi l ti It is a Ward equivalent with additional reactive support at the boundary buses such that its

ti i l t th t f threactive response is close to that of theWard-PV equivalent.

The reactive support in the extended Ward is derived so that the incremental response p(linearized response from the base case) for the reactive power flows is almost the same as pthat from the Ward-PV equivalent.

Incremental form of Decoupled Load Flow:T

PPP

TQVP

QQ

VPB

VP

'..... ,'

1

1

VQ

VQVB

VQ

'..... ,''

1

1

Consider the decoupled load flow: jiij ijjii BVVP

I t l f

ij ijjiiiii

j

BVVBVQ 2

Incremental form:

BVVP jiijj

ijii

: then,1Set

:or

V

BVVP

j

jiijij

ji

i

flow) load (DC '

BVP

On the other hand:BVBVV

Qijjiii

i 2

VQ

BVVQV

ijij

i

iji

ji

1

BVQ

V

BVVBV

QV

ijj

ii

ijij

i

jii

i

ii

ji1

1

2

:so

VQ

VQ

VQwhereV

VQ

QAs2

1

1

1

,

VVV

VVQQ 112

1

1

BBB

V

BF

VBVVQ

VQ

''

1

2''

''

BBB

BBB

ijiji

ijij

ijijiiiiBFor

shunts0 2

2 :''

BGY iiiiii jBB i

ijij

B ii

02

'

jiBB ijij ''B can be obtained from B by deleting rows & columns corresponding to PV buses, doubling p g gthe shunts.

* * * * * * *From Eq. (+), if the rows and columns corresponding to the external PV buses are deleted, then the decoupled reactive power flowis given by:

bQ

b

b

bbiibbWV

VVBBB

''''''

i

iiiiib

V

V QBB ''''

The reactive response of the Ward-PV Equivalent to the changes in boundary busEquivalent to the changes in boundary bus voltage isbV

WV VBVQ ''

001b

b

VV

(*)On the other hand if one starts with the original

bWVbb VBVQ 0 (*)

networks and performs the Ward Equivalent, the decoupled reactive power flow would be

Q '''' ''

'' ''

b

ibbW b b b i

QVVB B B

V QB B

The reactive response of the Ward equivalent is

i iib i ii

V QB BV

given by:

W '' bWbWb VBVQ ''

By equations (*) & () If the Ward Equivalent isBy equations ( ) & (). If the Ward Equivalent is desired to have the same reactive response as the Ward-PV equivalent reactive injectionsthe Ward PV equivalent, reactive injections should be of the amount:

~ (#) bWWVbb

VBBVQ ^

''''(#)

* * * * * * *B^

An approximation to (#) will be derived that can be implemented in a power flow program Thebe implemented in a power flow program. The approach is to construct a network the

^

corresponding B matrix of which is .B

Consider the k-th component of , i.e.,b

Q^

~

mkm

kmkk

BBQ

VVkkVk ^

^

kmkm

kmk BB VVVk^^

kBBB kkk ^^^

:Where

....(a)

I th d th di t k

km kmBBB kkk

In other words, the corresponding network has a shunt given by . kB

^

Now suppose wvBB '' and wkmw kmwv BBBB

''

:Then

and

bmkm

kvj

wvkj

wvkm

wvkk BBBB

(shunt)

:Then

bmkm

kwvkm

wkk BBB

(shunt)

wkkwvkk BBkkB^

:Therefore

vj

wvkj

kkkk

B

BBkkB

wvkjk

wkk

wvkkkm

Bkk

BBVV

BB^^

:so , and :yPracticall

vj

kjk kkBB

By the above derivation, an approximate formula is obtained:formula is obtained:

kkkkVV BQ

^~

~~Bkj

Ward-PVkB

^

boundary external~

j

Bkj

j

kjk BB^

boundary vj

~kB

^2/

^

kBk2/2

2/

^~

^2

~

kkkk

kkk

BQBQ

VV

V

~~

~

The construction of an extended Wardequivalent is summarized below:equivalent is summarized below:

Obt i W d E i l t f th t l1. Obtain a Ward Equivalent of the external system.

2. Start again from the original system. Ground all external PV buses. Apply Gaussian Elimination on the Bus Admittance Matrix Y to eliminate all external buses to obtain the equivalent shunts at the boundary buses, which are the admittances y

kBj^

3. Augment the Ward Equivalent by inserting a shunt at each boundary bus.

^

kBj y

The extended Ward equivalent has been

2j

The extended Ward equivalent has been found to give accurate results for contingency evaluationcontingency evaluation.

Further InformationF F W d A M ti lli C iti l i fF. F. Wu and A. Monticelli, Critical review of external network modeling for on-line security

l i I t J l f El t i l P danalysis, Int. Journal of Electrical Power and Energy Systems, Oct. 1983, pp. 222-235.