análisis rayo sobretensiones por emtp

7/26/2019 Anlisis Rayo Sobretensiones Por EMTP

1/4

AN IMPR OVED ALGORITHM OF BEWEL Y'S LATTIC E DIAGRAM METHOD

FOR THE SOLUTION OF ELECTROMAGNETIC TRANSIENTS

Abdul hal em A. Naz i

Electrical Engineering Department

C o l l e g e o f E n g i n e e r i n g , K i n g S a u d U n i v e r s i t y

R i y a d h , S a u d i A r a b i a

A B S T R A C T

A study of e lect roma gnet ic t ra ns ie nt s (EMT),

1n power sys tems 1s es se nt ia l fo r the prop er

des ign o f i nsu la to rs and p ro tec t i ve dev i ces .

Among the various avai lable methods of solving

EMT, Bew ely 's la tt ic e diagram method is simple

In concept and suitable for academic purposes

whe.-e the objective 1s to demonstrate the

phenomenon of traveling wave and the voltage

bu i l d- up due to ref le cte d and refr act ed waves .

The accuracy of Bewely's method depends on the

sel ect io n of an arb i t r ary t ime In ter va l At to

account for a l l the ref lec t io ns and re f r ac t i on s.

The present paper de sc ri be s an Improved

al gor i t hm which does not requ ire an ar bi tr ar y

se le ct io n of time int erv al A t. Here the exact

time Int er va l for a wave to tr ave l between two

nodes is cons idered for the ca lcu lat ion of

subsequent re f l ec t i on s. Only s i ng le phase

calculat ion Is considered, but the method could be

adopted for three phase ca lcu lat ion.

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

Surges in power systems are initiated due to

severa l reasons such as swi tch ing ope ra t io ns,

fa u l t s , l i gh tn in g , e tc . The d i s t r i b u te d na ture

of t ra nsm is s io n l i ne parameters causes

el ectrom agnet ic waves to trav el thr oug hou t the

sy st em. These waves may le ad to se ve r

over vol t ages and may cause ins ul at io n fa i l ur e.

For proper des i gn of t r an sm is s i on l i ne s

and the ir pro te ct i on schemes i t is requ ired to

stud y the eff ect of sur ges In power sy st em s. Two

methods for graph ical so lu t i on of tr av el in g waves

in the 19 20 's and 19 30' s were in tr od uce d. These

are . the Bew el y' s la tt ic e diagram techni que and

Ber ger on ' s method ( 6 ] ,

Lat t ice d iagram method prov ides a p ic to r ia l

des cr ip t io n of the ef fect o f inc i den t ,

re f l ec te d, and refract ed waves. In, 1 ementat l on

of the method on di gi ta l computer was re por te d

in 1961 for the f i rs t t ime [2 ]. A vol ta ge surgeor ig in at in g at some poin t in the system

propagates over t r an smi ss i on l i ne s or cables

througho ut the sys tem . At po in ts of

d i sc on t i nu i t y , surge Impedance determines the

po rt io n of the wave which r ef le ct s back or Is

transmitted to other parts of the system. Refl ec t i on and t ransmiss ion Coe f f i c i en ts

are cal cul ate d based on the val ues of syste m

parameters . The exp re ssi on s for these

coe ff ic ie nt s are give n in the appendix [ 1, 2, 5, 6 ] .

It takes the wave certain travelling time to reach

a ter mina tio n po in t. The surg e volt age level at

any time is the sum of all in ci de nt and re fl ec te d

waves at the point of discontinlty up to that

Ins tan t of t ime. Cal cu la t i on of the volta ges at

each node requires a systematic bookeeping for all

In cid ent , re f l ec te d, and refr acted waves. For

dig i ta l computer so lu t i on thi s cold be handled

using the same concept of Bewley's lattice

D i a g r a m .

B E W E L Y ' S L A T T I C E D I A G R A M

Th is is a gr ap hi ca l method to determine the

voltag e at any point in a tran smi ss io n syste m.

Two axes are es ta bl is he d. A horizontal one

scaled in distance along the system, and a

vert ical one scale d in t ime. Lines ind ica t i ng

the passage of surges are drawn such that their

slo pes give the t ime corr espo ndin g to dist ance

tra vel ed. At each t r an s i t i on point , the ref lected

and tran smit ted waves are obtai ned by mul ti

plying the incident wave by the appropriate

re f l ec t i on and re f rac t i o n coe f f i c i en ts .

Graphical so lut ion l i mi ts the appl icat ion of

la tt ic e method to simple radial syst ems. Digi tal

computer so lu ti on makes it po ss ib le to apply the

same concept for complicated transmission systems.

Al so dif f erent l i ne ter mina tio n as well as surges

of different shapes could be handled very

ef fe ct iv el y. Barthol d and Car ter [ 2 ] introduc ed a

method of dig i ta l computer so lu t i on for tra vel ing

waves in power sys tem. Fol l owin g sect ion descr ibe s

an improved method for di gi ta l computer sol ut io n

of latt ice diagram method.

I M PR O V E D D I G I T A L S O L U T I O N O F B E W E L Y ' S

L A T T I C E D I A G R A M M E T H O D

The main con cep t of the method l i e s in

considering surge as it eminates from a node over

a tra nsm iss ion l i ne . The source of the surge

could be due to ref lect ion, refract ion, or due to

any switching operation at the node.

491


2/4

The reflected wave wil l

new wave t ra ve l i ng over a

d i rec t i on to the o r ig ina l

re f rac ted wave w i l l i n i t i a te

l i ne s connected to the node

each physi cal l i ne connec ting

represented as two oriented l ines

be considered as a

l i ne w i th oppos i te

one, whereas the

new waves over al 1

To ach ieve th is ,

two nodes is

The fi rst one

Block No. 4

is ori ent ed and the second one Is ori ent ed

Here the fi r s t index is the From- node and

the second index is the To-node. To i l lustrate

t h i s , cons ider the sys tem shown in F i g , ( l ) .Or ig in al surge V is moving from node ( !) towards

node (2) ever li ne LI ( 1 , 2 ) . By the time th is

sur ge reaches node ( ? ) , new waves wil l be

in i t ia te d and propagate over a l l l in es in c ide nt to

th is node. These are l i ne s 1.2(2 ,3), 1.3(2, 4),and

L 5 ( 2 . 1 ) .

Figur e ( 1) : Sample 4- Li ne System

The above mentioned convention enables the

pro pos ed net hod to han dle any s yst em no mat ter how

complex it may be in the same way. The only

requirement is to monitor the surge on a given

l i n e till it reaches the o the r end. The effect ofgenera ted s u r g e s will then be considered as new

sur ges i n it ia ted at that node. Subsequent steps

assume the ef fec t of each newly gen er at ed sur ge in

any l i ne upon the l i n es c onne cte d to the remote

node onl y.

The voltage build-up at any node at a given time

is then obt ain ed by summing up all vo lt ag e

maan I tudes to the desired time at that node.

A block dianram of the pr< ?sed

t.ion is shown in F i g . (2) where :

com put e r sol u-

Block No. 5

Block No. 6

Determine the set [ L ] of al l l i ne s

In the system wit h waves I ni ti at ed

at the From-node at the same time.

For exampl e, at t2 , ther e are

th re e 1 i nes [ L2 ,L 3 ,L4 ] wi th waves

Ini t iat ing at the same t ime

as shown In Fig (3).

Rank the set of li ne s ( L I Ojta ined

Block No.4 in ascending order of

trav el time . The orde r of theth ree l i nes is [ L2 .L 5. L3 ] .

For each l i n e K ( i . j ) , K C L , (where

(i ) is the From-node and (j ) i s the

To-node for l i ne K) determine the

l i n es [ C] which are connect ed t o

node (j ). For example, L2 (2 ,3 ) is

connected to l i ne s L4 (3 ,4 ) and

L6 (3 ,2 ) . When the inc ide nt wave

on l i n e K reac hes node (j ) 1t will

in i t ia te new waves on al l l in es

connected to it.

RLAD INPUT DA

CALCULATE TFJWKL TIKE FOR E A C H LIN E

B lo ck , ]

Mo ck l2

CALCULATE T,JE REFLECTION AND

RETRACTION COEFFICIENTS

S C A N ALL LI NES [L] WITH WAVE?

INI TIA TED AT THE SAME INSTA NT

RANK THE LIN ES [ Ll IN ASCENDIN G

ORDER Of WAVE TRAVEL TIHE

FOP E A C H LIN E M i , J ] , ( K . T J M F

1- Calcu lat e the tot al vol tage at each node

in the system |

2- Save the naxlnuirr. vo l ta ge and th~ time at j

which ocr ure d [

Fi g. ( 2 ) : Program Block Diagram

SURGE WAVE SHAPES

P.l ock No.l : Input l in e co ns ta nt s and system pa

rameters are read.

m ock No. 2 : Ca lc ul at e wave tr ave l time for each

1 i ne seg men t.

Block No. 3 : Cal cu l ate the re f l ec t i on and ref ra

c t i on co e f f i c i en ts for each l ine .

Or ien ted l i ne no ta t i on i s cons ide

red for th is purpose.

The method Is arranged to accomodate step

wav e-f onr .s. Any othe r wave-for m is handled by

decomposing it Into several step wave-forms ofdi ff er en t hei ght and with app rop ria te t ime drl ay

between different segments. The simulation of

li gh tn in g wave-form by sequence of step funct ion s

(r ect ang ul ar waves) is shown in Fig . (A2) of the

appendl x.

492


3/4

F ig u re (3 ) : Mo di fi ed La t t i c e uiagran. for the S,-!ir.}>le System of Figu re (1)

LIN E TERM I NAT I ON

Line terminat ion with f ixed value parameters

behave di f f ere nt than t ra nsm iss io n l i n e s . These

req uir e speci al t reatment for prope r es t im at io n of

the i r surge impedance val ue. Af ter es t i mat i ng

the wave trave l t im e, the surg e impeo-rnce co ul d be

ca lc ul at ed . The travel t ime is est im ated to have

a value equal to 10% of the sma ll es t trav el t ime

of any l i ne in the sys te m. Too small tra vel t i me

caus es ove rf lo w proble m, and too b1 g val ue ca uses

si za bl e er ro rs . The 10% value 1s obt ain ed af te r

s ev era l t r a i l s o f d i f f e r e n t t e r m ina t i o n .

E X A M P L E S A N D T E S T C A S E S

EXAMPLE No .l : A two l i n e s syste m ter min ate d by a

tra nsf orm er 1s shown in F i g . (4 ) . A step

function surge of 500 KV is moving towards node

(2 ). The tra nsf orm er 1s repr esen ted hy an

induc tor and a capac i tor connec ted in paral le l .

Af te r 0. 320 msec the vo lt ag e at the tr an sfo rme r

node reached a value of 662.529 KV. If this

voltage 1s greater than the transformer rt iaximum

ins u la t i o n w i t hs tand l ev e l , a h r eak - down cou ld

re su l t and the tra nsf orm er wi l l be damaaed.

500 KV

}-*

r-

L l

-CD

L3 u

Example No.2: Fig(5) shows l ightning surge with

discharge current of 20 KA at the middle of l ine

L I . A vol tag e surg e of magnitude 3742 KV

(0 .5 *I Z, ) wi l l be produced and propagate on both

di re ct io ns of the l i n e . After 1.11 msec the

vo lt ag e at node ( 2) re ache d a valu e of 75 75 . 328

KV, where as the vo lt ag e at node (1) reach ed a

value of 7667 .356 KV aft er 2.0 2 msec. I t is

usefu l to know that at t ime 2. 5 mse c, the vo lta ge s

at no de s( l ) , ( 2) and (3) were 185.184 KV, 93. 728

KV, and zero re sp ec t i ve ly . The reduct ion in the

voltage is due to reflected waves of opposite

polarity to the Incident wave. Lower part of F1g.

(5) shows the lattice diagram up to a time of 2.5

m s e c .SUMMARY

Ap pl ica t i on of la t t ic e diagram in step s equal to

the actual time of tra vel between two nodes is

ac hi ev ed . The eff ect of all gene rated waves is

counted with no possibi l i ty of missing any

ref lected or refracted waves. Lightning wave-form

is handled as sequence of step function waves. A

computer program is develop ed for the implementa

ti on of the prop osed meth od. The progr am is used

to compute vol ta ge bu il d- up for several tes t

sys te ms . It is shown that the proposed algo rit hm

is simple in concept-, accurate and can be very

ef fe ct iv e tool in teac hing the subject of EMT.

ACKNOWLEDGMENT

Author l i ke s to thank Dr. N. Ma lik for revi ewin g

the manuscr ipts and valuable sugges t ions .

R E F E R E N C E S

Fi gu re (4 ): System Fxampl e No .l L.V. Bewely, Travel ing waves on transmission

systems. New York, Dover 1963.

493


4/4

L 2

1 ^

20 KA

-

análisis rayo sobretensiones por emtp

Documents