Dimensioning of nearby substations interconnected ground system

IEEE Transactions on Power Delivery, Vol. 3, No. 4, October 1988 1605

DIMENSIONING OF NEARBY SUBSTATIONS INTERCONNECTED GROUND SYSTEM

S .T. SOBRAL VASCO S .COSTA MARCF.LO S. CAMPOS Senior Member Senior Member BRAULIO GOLDMAN IESA-Internacional de Engenharia S.A. ELETROBRAS Rio de Janeiro BRASIL BRASIL Rio de Janeiro

Non-Members Rio de Janeiro IESA-Internacional de Engenharia S.A.

BRASIL

D. MLJKHEDKAR Fellow Member ECOLE POLYTECHNIQUE Mont r ea1 CANADA

ABSTRACT - This paper deals with the ground mat dimensioning of two or more neighbour interconnected substations, a situation that is very common in the Elec- trical Industry.

The paper recalls that the external ground circuits connected to the ground grid of each substation can drastically reduce the percentage of total ground current injected into the soil through the mat (from 40% up to 2% of the total fault current).

The paper presents a set of specific calculation procedures to deal with nearby interconnected ground mats. These procedures correspond to a particular il- lustration of the general "Decoupled Method" [ 3,4,5], showing how to apply its 8 sequencial steps to solve this type of circuit.

The paper shows that the electric neighbourhood of nearby subs tations depends on the"Space Cons tant"(or "Characteristic Length")of the ground circuits interconnecting them such as transmission line ground-wires, power cable sheaths, etc.

This paper complements also Ref. p.41, intro- ducing the complete derivation of usefull expressions used to solve lumped parameter ladder circuits of any size (from one pi to an infinite number of pis). The derivation of these expressions also used in p,q were not included in these References due to lack of space.

In the paper it is also shown a simple procedure to determine the suitable number of ACSR ground-wire spans near a substation necessary to allow a reduction of the ground grid conductor extension. 1. INTRODUCTION

The existence of two or more power substations only a few kilometers apart interconnected by over- head transmission lines or power cables, is very like- ly to occur in the Electrical Industry. For instance: the several substations of a transmission network feed- ing a town; the generation station and the switchyard of a power plant ; the step-dotm substation of an industry and the corresponding utility substation, etc.

The ground mats of these substations are generally interconnected by external ground circuits, such as ground-wires of the over-head transmission lines, power cables sheaths and additional underground interconnecting circuits.

It is well known that external ground circuits connected to a ground mat can drastically modify the copper mat conductor spacing. For instance, when ACSR ground-wires are used in some transmission line spans near the substation,possibly only 12-30% of the total fault current is injected into the soil through the local ground mat [4,57. When a substation is fed exclusively by power cables, this percentage

87 SN 527-5 A paper recommended and approved by the IEEE Substations Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1987 Summer Meeting, San Francisco, California, July 12 - 17, 1987. Manuscript submitted February 2, 1987; made available for printing May 11, 1987.

can be as low as 2-4% as shown in Eo?. Hence, before the mat design, it is mandatory to determine carefully the ground current injection i n t o the s o i l , which is shared among the ground mats and the external grounding circuits.

For convenience, lumped parameter ladder circuits, composed of several cascaded pis are generally used to represent the underground circuits interconnecting the mats, as well as to represent the over- head ground-wire circuits and its grounding points through the towers.

Each pi of this ladder circuit is generally coupled with the phases of the transmission lines and cables. Hence, the complete circuit necessary to represent the interconnected ground circuits of nearby substation has generally several hundreds or even thousands of circuit elements.

During the last decade several methods have been proposed to solve this type of circuit, most of them u- sing special matricial techni ues. However,theoretical and practical considerations 19,3,4j seem to show that the relative error propagation can be a problem when solving a large complex matri-x. -

The Decoupled Method L3,4J avoids this problem through eight sequencial "steps" which allow the prepa- rationof a decoupled and reduced circuit which can be solved with minimum relative error propagation. The fourth "step" of that Method is called the "decoupling technique" and states that all the magnetic couplings between a ladder circuit and the supply circuit phases can be replaced exactly by current sources applied between specific points of t_he circuit and the remote ground (see Annex I and L3,4]).

Since the late 1960's many authors (Endrenyi,Po- ter, Finch,Johnson,Sebo, Fesonen, Dawalibi, Mukhedkar, [l, 2, 8, 9 , 3) etc.), have presented methods for the solution of ladder circuits. The approach generally u- sed by these authors has been to replace the lumped parameter ladder circuit by an equivalent distributed parameter ladder circuit solving it by differential e- quations. This process leads to good results provid- ing the number of-cascaded pis is large enough. How- ever, an important discrepancy would appear in the results when applying this method to the very common case of a small ladder circuit that replaces for instance 3 or 4 ground-wire spans interconnectingtwoneighbhouring substations. The paper shows that the exact solution of a uniform ladder circuit of any size (from one pi up to infinite number of pis) can be carried out using very simple formulae developed directly for lumped parameter ladder circuits.

The derivation of these formulae were not presented in 3,4 due to lack of space, but is now fully presented in Annex 11. The simultaneous utilization of the Decoupled Method and of the Lumped Parameter Ladder Circuit Techniques allows even very complex grounding systems to be solved by means of relatively simple mi- crocomputer programs with minimum storage area requirements.

The paper shows also a simple procedure which allows the determination of the suitable number of ACSR ground-wires to control all ground potential problems at a substation.

0885-8977/88/1ooO-1605$01 .WO 1988 IEEE

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phases of both lines and with the corresponding imped- 2. SUMMARY OF SUBJECTS ance (2~1) of the ground-wire of the other line.

sented in the following order: The ladder circuit of the (g.w) of the line be- The subjects dealt with in this paper are pre-

0 Section 3-presents the the basic description of a sample system selected to illustrate the application of the Decoupled Method for a common case of interconnected nearby substations.

Section 4 - describe the application of the 8 steps of the method to solve the system. 0 Section 5 - shows that the final circuit obtained after these 8 steps can be directly modified in order to represent fault positions in all the critical points or to introduce local circuit modifications.

0 Section 6 - deals with the concept of remote or neighbour ground installations and how to use this concept to carry out useful "a priori" simplifications of the ground system. 0 Annex I - reminds the basic concepts of the "Decou- pled Method" and of the "self-neutralized" current, introduced in E,&]. 0 Annex I1 - present the development of the formulae used for the manipulation of lumped parameters ladder circuits of any size (from one pi up to an infinite number of pis). The derivations of these formulae was nct included in p,g for lack of space. 3. SAMPLE SYSTEM AND OBJECTIVES

This section describes the sample system selected to illustrate the applicationof the Decoupled Method r3, 47 to the design of interconnected ground &stems of nearby substations. The objectives of the calculation are also specified.

I c.

Fig. 1 - Sample System Configuration

3.1 ILLUSTRATIVE SYSTEM Fig. 1 shows a part of a transmission system in-

cluding 3 interconnected substations (SE-A) , (SE-B) and (SE-C). Substations (SE-A) and(SE-B)are lOOkm apart and are interconnected by two parallel transmission lines. Substations (SE-B) and (SE-C) are only 10 km apart. The ground mat resistances of the 3 substations are (X1), (Rz), (R3),in Ohms. The ground mats of these 3 substations are interconnected by lumped parameter ladder circuits representing the steel ground-wires (g.w) of the transmission lines. The ladder circuit of the(g.w)s of the line between (SE-A) and (SE-B) is composed by 300 pis,with longitudinal impedances (ZL~), in Ohms, and transverse impedances (ZT1), in Ohms. Each impedance (2~1) is submitted to mutual couplings with the

tween (SE-B) and (SE-C) is composed of 30 pis with longitudinal impedances (ZL~), in Ohms, and transverse impedances (2T2), in Ohms. Each impedance (2~2) is submitted to mutual couplings with the phases of the corresponding transmission line. The total number of circuit elements is 3896 as shown in Fig. 1. It should be observed that a similar circuit would result if (SE-B) and (SE-C) were interconnected by a power cable LlO]. In this case the cable sheaths would be represented by a similar ladder circuit coupled with its phases.

3.2 CALCULATION OBJECTIVES

It is assumed that the ground mat of (SE-B) is being dimensioned. It is well know that in certain cases the total amount of current injected into the ground through the mat is only a small percentageofthe total fault current. Hence it is mandatory to determine carefully the fault current distribution between the mat and the external ground circuit, for critical fault positions. The maximum value of current injected into the soil can be determined thereafter. Based on this value, the copper conductor spacing necessary to control ground potentials can be determined, using a computer program 01 approximated formulae.The modification of the mat conductor spacing can control the following ground potentials: "step", "touch" and "mesh" potentials; transfered potentials to nearby or remote substations; gradient produced in the soil near the mat.

3.3 COMPLEMENTARY OBJECTIVE - IMPROVEMENT OF THE MAT DESIGN

Some utilities adopted the use of ACSR ground- -wires on the initial line spans near the substation (followed by steel ground-wires in the remainder o f the line). Originally this procedure was relatedto lightn- ing performance. However it was soon verified that it could drastically reduce the amount of fault current injected into the soil through the mat. In the measurements reported in [ 5 ]this amount is only 12.8% of the fault current for the worst fault position. Hence, the mat design can be improved by repeating the calculation of ground mat conductor spacing while increasing by one the number of spans with ACSR ground-wire. Generally after a certain number of ACSR spans, the increase of cost with a new ACSR span is no longer compensated by a significant decrease of conductor length or a reduction of ground potentials. The method allows a great simplification of the calculations because all the spans to be modified can be kept in the final equivalent circuit (see towers (3) to (11) near (SE-B) in Fig. 1).

4 . CALCULATION PROCEDURES

In this section the 8 "steps" described in p.47 are applied to the sample system in order to illustrate the simplicity and strength of the method.

4.1 STEP 1 - "MONITORED POINTS" SELECTION the final

equivalent circuit and includes points from(1)up to (12).

4.1.1 Determination of the Number of ACSR Ground- Wire

The"monitored points"wil1 be kept in

Spans

Points ( 3 ) , ( 4 ) , (5), (6), (7 ) , (8), ( 9 1 , (10). -

(11) are selected in order to make easier the modification in sequence of the ground-wire spans froni steel to ACSR conductors, near (SE-B) span by span.

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4 .4 STEP 4 - "DECOUPLING TECHNIQUE" F ig . 3 shows a l s o t h a t a l l t h e coupl ings be-

tween the ground c i r c u i t s and supply c i r c u i t phases were replaced by 4 c u r r e n t s sou rces , connected between the t e rmina l s of t h e "uniform sec t ion" . This procedure has been c a l l e d [ 3,4,51 "decoupling" technique". The a p p l i c a t i o n of t h i s technique reduces the number of c i r c u i t elements from 3096 t o only 1560 c i r c u i t e l e - ments.

It should be observed, however , that 300 magnetic coupl ings between the ground-wires of the p a r a l l e l l i n e s remained ye t i n the c i r c u i t .

4.5 STEP 5 - REPLACEMENT OF NEUTRAL AND FAULT CUR- RENT BY ADDITIONAL SOURCES F ig . 3 shows a l s o t h a t t he n e u t r a l c u r r e n t s ( 1 )

and (IN2) (f lowing from t h e t r ans fo rmer n e u t r a l s Nko t h e mats) and t h e f a u l t c u r r e n t (Ic) determined i n s t e p 2 were a l s o r ep laced by current-sources (with one terminal connected t o remote ground),

t

4.1.2 C r i t i c a l Fau l t P o s i t i o n s Producing Maximum Po- t e n t i a l T rans fe r Maximum p o t e n t i a l t r a n s f e r between s u b s t a t i o n s

(SE-B) and (SE-A)occurs f o r f a u l t s t o ground a t one of t hese substat ions.The same f o r maximum p o t e n t i a l t r ans - f e r between (SE-B) and (SE-C). Hence, i t i s necessary t o apply f a u l t s a t t he 3 s u b s t a t i o n s ( p o i n t s (l), (2 ) , (12)) *

4.1.3 C r i t i c a l Fau l t P o s i t i o n s Producing Maximum Mat Current

The f a u l t p o s i t i o n s which produce the maximum cur ren t flowing between (SE-B) ground mat ..nd there- mote ground occurs f o r one of t hese p o s i t i o n s 1 3 1 : a ) Fau l t t o ground a t (SE-B);

b) n e s , a t a d i s t a n c e 6 (CE1) from (SE-B) . (see Fig. 1): , ,

F a u l t t o ground along one of t he two 100 km li-

The d i s t ance (CE1) can be computed as fo l lows

-zL1 /$ + ZL1.ZT1 K = ~ 'T1

2 'T1"E 1 ZL1 = - +

K1 = Module of the complex (K)

S = Ladder c i r c u i t r e g u l a r span = 0.33 km

, i n km - s in (IC1)

C) F a u l t t o ground along t h e 10 km l i n e , a t a d i s - tance 6 (CE2) from (SE-B).

The c a l c u l a t i o n of (CE2) uses t h e same formulae above, and t h e l adde r parameters ( Z L ~ ) , ( Z T ~ ) . The re- s u l t i s CE2 : 3.3 km. Hence 6 (CE2) = 19.8km.Therefore a s t h e l i n e has on ly l o b , t h e c r i t i c a l f a u l t cond i t ion corresponds t o a f a u l t t o ground a t (SE-C).

Remark: I n o rde r t o make s impler t h e example, l e t us assume t h a t cond i t ion (b) could be "a p r i o r i " e l imina ted ( t h e s h o r t - c i r c u i t c a l c u l a t i o n men- t i oned i n s t e p 2 showed t h a t t h e s h o r t - c i r c u i t c o n t r i b u t i o n i n these p o i n t s could be d i s r e - garded when compared with case ( c ) ) . This e l i - minat ion avoids the need of cons ide r ing mon- i t o r e d p o i n t s a long t h e 100 km l i n e s .

4.2 STEP 2 - ZERO SEQUENCE DISTRIBUTION ALONG THE PHASES AND TRANSFORMER NEUTRALS Fig. 2 shows t h e r e s u l t s of a s h o r t - c i r c u i t

s tudy c a r r i e d ou t u s i n g a convent ional s h o r t - c i r c u i t program, cons ide r ing a f a u l t a t (SE-C)(or "monitored po in t " number(l2)) .The z e r o sequence c u r r e n t s (11) and (12) are f lowing a long t h e t r ansmiss ion l i n e phases.The ze ro sequence c u r r e n t s ( 1 ~ 1 ) and ( I ~ 2 ) a r e f lowing along t h e t ransformer n e u t r a l s . The c u r r e n t (IC) i s i n j e c t e d a t t he f a u l t p o i n t i n (SE-C).

It i s necessa ry t o p repa re a d d i t i o n a l sho r t - c i r - c u i t c a s e s , f o r f a u l t s a t (SE-A) and (SE-B), which a r e the o t h e r c r i t i c a l f a u l t p o s i t i o n s .

-

Fig. 2 - Zero Sequence D i s t r i b u t i o n Along t h e Phases

4.3 STEP 3 - IDENTIFICATION OF THE UNIFORM SECTIONS Fig. 3 shows t h a t 4 "uniform s e c t i o n s " were i n -

d e n t i f i e d along t h e ground c i r c u i t s , between p o i n t s

and Transformer Neu t ra l s

(1-21, (-21, (2-111, (11-12).

Fig. 3 - Sample System Configurat ion Af te r S t ep no.5

4.6 STEP 6 - REDUCTION OF PARALLEL SECTIONS The two 100 km l i n e s a r e p a r a l l e l and composed

by s i m i l a r l adde r c i r c u i t s . There a r e no c r i t i c a l f a u l t p o s i t i o n o r o the r ' hon i to red points"a1ong these l i n e s . Hence t h e l a d d e r c i r c u i t s r ep resen t ing the two ground- wires of t h i s l i n e s a r e " p a r a l l e l s e c t i o n s " and can b e reduced t o a s i n g l e equ iva len t ladder c i r c u i t , a s shown i n Fig.4. k6th t h i s procedure the numbers of c i r c u i t elemepts has been reduced from 1560 t o 661.

- r m

UmCiruDitmt 331 TMWSK 31 rorii 661

Fig. 4- Sample System Configurat ion Af te r Step 6

4.7 STEP 7 - REPLACEMENT OF LADDER CIRCUITS BY EQUIVALENT PIS Fig. 4 shows t h a t between t h e "monitored po in t s "

(1-2)and(ll-12) t h e r e a r e two ladder c i r c u i t . With sev- e r a l p i s each. Each one of t hese l adde r c i r c u i t s can be reduced t o an e x a c t equ iva len t p i using t h e s imple formulae shown i n i t e m 11-3 of Annex 11.

For convenience,one keep the l adde r p o i n t s a t r i g h t of (1) and a t l e f t of (2) and then the impedances (AI) and (B1) of t he equ iva len t p i , a s shown i n Fig. 5 . One keep a l so t h e ladder p o i n t s a t r i g h t of (11) and a t l e f t of (12) and then compute the i m - pedances (A2) and (B2) of t h e equ iva len t p i ( s e e F ig .5 ) .

compute

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23 4

By meanrof star-mesh t r ans fo rma t ion the termi- MONITORED POlHTS 12

Fig. 5 - Replacement of Ladder C i r c u i t s by Equivalent IMPEDllNCES CURRENT SOURCES P i s (A1-B1) and (A2-B2)

n a l p o i n t s of each equ iva len t p i can be e l imina ted , l ead ing t o the equ iva len t c i r c u i t shown i n F i g . 6, Fig. 8 - Final Decoupled and Reduced C i rcu i t ,Equ iva len t

t o That Shown i n F i g -

Fig. 6 - Addi t iona l Reduction o f t h e Eqquivalent P i s Between (1-2) and (11-12)

The c i r c u i t shown i n Fig. 6 h a s 3 c u r r e n t sources connected t o remote ground and 3 o t h e r cur- r e n t sources connected between two p o i n t s of t h e c i r - c u i t . For convenience each one of t hese 3 las t c u r r e n t sources a r e r ep laced by 2 c u r r e n t sources . For in- s t ance t h e source (-1~~') connected between p o i n t s (1) and (2) i s r ep laced by t h e sou rces (+IR1')connected between p o i n t (1) and t h e remote ground and by (-IR1') connected between p o i n t (2) and t h e r e m t e ground. The r e s u l t i n g conf igu ra t ion i s shown i n Fig. 7.

Fig. 7.- Manipulation of t h e Current Sources Connected Between Two P o i n t s of t h e C i r c u i t

F i n a l l y Fig. 8 shows the f i n a l conf igu ra t ion of a decoupled and reduced c i r c u i t w i t h 12 "monitored po in t s " , 23 impedances and 4 c u r r e n t sou rces which i s e x a c t l y equ iva len t t o t h e o r i g i n a l c i r c u i t shown i n Fig. 1 wi th 3096 c i r c u i t e lement .

4.8 STEP 8 - MATRICIAL SOLUTION OF THE DECOUPLED AND REDUCED EQUIVALENT CIRCUIT The c i r c u i t manipulat ion c a r r i e d o u t a long the

former 7 "s teps" i s based on b a s i c c i r c u i t theory and involves a minimum amount of numerical ope ra t ions (and hence minimum r e l a t i v e e r r o r propagat ion) . Using c l a s - s i c a l ma t r ix procedures t h e vo l t ages of t he 12 "mon- i t o r e d po in t s " shown i n F i g . 8 can be determined with minimum r e l a t i v e e r r o r propagat ion. For i n s t a n c e , t he vo l t age (V2) i s t h e ground p o t e n t i a l r i s e of t h e mon- i t o r e d p o i n t (2) with r e fe rence t o remote ground,dur- i ng a f a u l t t o ground a t p o i n t (12 ) .

The c u r r e n t I R ~ flowing t o ground through t h e mat of (SE-B) is:

1 ~ 2 = V2/R2 i n Amps

One can now determine the ground g r i d conductor spacing a t (SE-B) necessary t o keep t h e ground poten- t i a l s bel low t h e t o l e r a b l e l i m i t s , when a c u r r e n t (IR2) i s i n j e c t e d i n t o ground through the m a t .

by s p e c i f i c computer programs (not d e a l t w i th i n t h i s paper) o r by means of approximate expres s ions ( f o r in- s t ance , IEEE-80 formulae) .

The de te rmina t ion of ground g r i d conductor c ros s s e c t i o n t o s a t i s f y thermal requirements must consider t he maximum amount of c u r r e n t f lowing through t h e c ros s s e c t i o n of t h e ground g r i d conductor.Hence t h e percentage of t h e f a u l t cu r ren t used t o compute the r equ i r ed ground g r i d conductor c ros s s e c t i o n can be s e v e r a l times b igge r than t h a t used t o compute the ground g r i d conductor spacing. The ground g r i d conductor c r o s s sec- t i o n can be determined us ing c l a s s i c a l IEEE-80 s t anda rd procedures .

A l l t he va lues of ground c u r r e n t ob ta ined by s o l u t i o n of t h e c i r c u i t shown i n F ig . 8 a r e f i n a l ones and need no a d d i t i o n a l c o r r e c t i o n s . However, t he lon- g i t u d i n a l c u r r e n t f lowing between two p o i n t s must be added t o t h e corresponding va lue of t h e c u r r e n t source b r i d g i n g these p o i n t s . For i n s t a n c e , t h e c u r r e n t flow- i n g between p o i n t s (3) and (4) ob ta ined by t h e s o l u t i o n of t h e c i r c u i t shown i n F i g , 8 must be added t o t h e c u r r e n t source b r i d g i n g p o i n t s ( 3 ) , ( 4 ) , a s shown i n F ig . 6.

The mat conductor spacing can b e determined

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The paper a l s o shows how t h e method can be used t o determine t h e s u i t a b l e number of ground-wire spans nea r t h e subs t a t ions i n which t h e s t e e l ground-wire i s t o be changed t o ACSR ground-wire i n o rde r t o improve t h e performance of t h e grounding system.

The paper in t roduces a simple process t o c l a s s i f y two ground i n s t a l l a t i o n s as"neighbour"or ''remote'!

The paper a l s o in t roduces the d e r i v a t i o n of i m - po r t an t express ions which a l low the manipulation of lumped parameter ladder c i r c u i t with-any number of p i s . This d e r i v a t i o n was not included i n L3,4] due t o l ack of space . The de r iva t ion of t h i s formulae i s based d i - r e c t l y i n lumped parameter ladder c i r c u i t , and no t a s usua l on d i s t r i b u t e d parameter c i r c u i t s . This approach i s no t very o f t e n found i n l i t e r a t u r e and may enhance the i n t e r e s t about t h e sub jec t . REFERENCES

1. J. Endrenyi "Analysis Cond i t i ons " I E E E Trans PAS-86 no. 10 , 1967 - pp 1274-1983

2. J. P o t t e r and S . Finch Theory of Netw0rk.s and Lines (Book) McGraw H i l l 1959.

3. S.T. Sobra l , V. F leury , J .R .Vi l la lba , D. Mukhe- dkar "Decoupled Method For Studying Large In te rconnec ted Ground Systems Using Microcomputers - P a r t I - Funda- men t a l s " I E E E Transac t ion Paper 86SM 455-0

4 . S.T. Sobra l , V. F leury , J . R . V i l l a l b a , D. Mukhe- dkar "Decoupled Method For Studying Large In te rconnec ted Ground Systems Using Microcomputers - P a r t I1 - U t i l i - z a t ion on the I t a i p u Ground System and Complementary Aspects" I E E E Transac t ion Paper 86SM 456-8

5 . S.T. Sobra l , C.A. Pe ixoto , D.Fernandes,D. Mukhe- dkar "Grounding Measurements a t t he I t a i p u Generation Com- p lex Using the Extended Eleck Method" I E E E Transac t ion Paper 86SM 457-6

6. S.T. Sobra l , C.A. Pe ixoto , D . Mukhedkar "Ground P o t e n t i a l D i s t r ibu t ion i n the Neighbourhood of It a i p u Generation Comp 1 ex " IEEE Transac t ion Paper no. 85SM 319-9

7. i. Pohl " Inf luence of High-Voltage Over-Head Lines on Covered Pipe l i n e s " CIGRE 326, 1966

8. W.C. Johnson "Transmission Lines and Networks" (Book) MacGraw H i l l , 1955

9. F. Dawalibi "Discussion of t he paper "Analysis of Grounding Sys- tems" - Vol. PAS 100 no. 3, March 1981.

10. S.T. Sobra l , J .O. Barbosa, J . V . C . Nunes, E . Chi- n e l l i , Y . S . Costa, J . Hisbe l lo Campos "Fault Appl ica t ion Transfer" i n Urban Step-Down Substa- t i ons - A P r a c t i c a l Example" Paper submitted t o I E E E

o f Transmission Tower P o t e n t i a l s During Fau l t

5. STUDY OF OTHER FAULT POSITIONS AND LOCAL MODIFI- CATIONS

The equiva len t c i r c u i t shown i n Fig. 8 was de- r ived assuming a f a u l t t o ground a t po in t (12). However the same impedances remain v a l i d t o f a u l t p o s i t i o n i n any o t h e r of t he 12 monitored poin ts .For each new f a u l t pos i t i on only the va lue of t h e cu r ren t sources w i l l change.

I f one want t o modify t h e ground-wire conductor between po in t s (3) and (4) t h i s modi f ica t ion can be in t roduced d i r e c t l y i n t h e c i r c u i t wi thout modi f ica t ion of any o the r c i r c u i t impedance. It i s a l s o necessary of course t o modify t h e va lue of t h e cu r ren t source ,br idg- ing the modified lonp i tud ina l impedance (because(p)wi l l change-Annex I ) . 6 . THE CONCEPT OF NEIGHBOUR SUBSTATIONS

Fig . 8 shows t h a t po in t s (1) and (2) a r e interconnected through the impedance(W2)of t h e equ iva len t p i (W1, W2), W3). F ig . 8 shows a l s o t h a t (11) and (12) a r e in te rconnec ted through the impedance (W5) of t he equiva len t p i (W4. W5, Wg).

Sec t ion 11 .3 of Annex I1 expla ins t h a t the ohmic va lue of (W5) is f i n i t e because the d i s t ance between po in t s (11) and (12) i s sma l l e r than 3 times the "Space Constant" (or "Charac te r i s t i c Length") of t he ladder c i r c u i t between po in t s (11) and (12). Hence the cur- r e n t i n j e c t e d i n t o one of these po in t s can modify, t he p o t e n t i a l a t t h e o t h e r . Therefore t h e ground-c i rcu i t po in t s (11) and (12) can be c a l l e d " e l e c t r i c a l l y neigh- bours".

On the o t h e r hand the ohmic va lue of (Wg) i s ex- tremely l a r g e because the d i s t ance between po in t s (1) and (2) i s l a r g e r than 3 t i m e s the"Space Constant"(3SC) of each ladder c i r c u i t between po in t s ( l ) and(2 ) . Hence t h e cu r ren t i n j e c t e d i n t o one of these po in t s can not modify the p o t e n t i a l a t the o t h e r . Therefore the ground c i r c u i t po in t s (1) and (2) can be ca l l ed " e l e c t r i c a l l y remote ' I .

This concept allows the following genera l iza- t i on : when the ob jec t ive i s t o dimension the ground mat of a subs t a t ion such as (SE-B), po in t (2), i t i s nec- e s sa ry t o r ep resen t only e x t e r n a l ground c i r c u i t s up t o a d i s t ance of 3"Space Constants" from (SE-B). A s t he d i s t ance between (SE-B) and (SE-C) i s smal le r than (3SC) a l l the po in t s between po in t s (2) and (12)must be represented i n the ground c i r c u i t . 0 n the o t h e r hand the d i s t ance between (SE-B) and (SE-A)is l a r g e r than(3SC). From the po in t of view of the ground system of (SE-B) a l l the impedances connected t o (SE-A) need not be represented because the c i r c u i t i s opened by the i n f i n i t e impedance (W5).

The impedance (ZE) r e l a t e d t o one ladder c i r c u i t between (1) and (2) i s in te rconnec ted between po in t (2 ) and the remote ground. The same app l i e s f o r t he o t h e r ladder c i r c u i t . Hence the impedance (ZE/?)connected between (2) and the remote ground can r ep lace t h e re - mainder of the c i r c u i t t o t he l e f t s i d e of ( 2 ) .

I t can be observed t h a t t h e ca l cu la t ion of (ZE) f o r each decoupled ladder c i r c u i t (using express ion shown i n 4.1.3 above)can a l low an "a p r i o r i " r educ t ion of t he ground c i r c u i t s i z e . This reduct ion was not used i n Sec t ion 3 because the complete r ep resen ta t ion of t he two p a r a l l e l l adder c i r c u i t s was use fu l t o i l l u s t r a t e s t e p no. 6 .

7 . CONCLUSIONS

This paper shows how t o apply t h e Decoupled Method [3,4] t o dimension t h e in te rconnec ted ground system of two o r more nearby subs t a t ion , a s i t u a t i o n t h a t occurs very o f t e n i n p r a c t i c e . The method a l lows t h e s o l u t i o n of complex ground c i r c u i t s wi th seve ra l hundreds o r thousands of c i r c u i t elements avoid ing the r i s k of severe r e l a t i v e e r r o r propagation. The calcu- l a t i o n can be c a r r i e d out us ing r e l a t ive lys implemic ro - computer programs wi th minimum s t o r a g e a r e a requirements.

po in t s

1610

ACKNOWLEDGENENTS

Authors would l i k e t o express t h e i r g ra t i . tude t o d i r e c t i o n of IESA, and ELETROBRAS f o r a l lowing t h e pub- l i c a t i o n of t h i s paper ,

Sobral and Mukhedkar wish t o express t h e i r debt of g r a t i t u d e t o Natura l Science and Engineering Research Council - (NSERQ - Canada) and Conselho Nacional de Pcs- qu i sa (CNPq-Brasil) f o r provid ing the funds f o r t he be- ginning of t h e i r coopera t ion .

Sobra l has an s p e c i a l debt of g r a t i t u d e t o Profes- s o r s J.S. Ol ive i r a , Mario Miragl ia , I r a n S. Pin to from SENAI, f o r t h e i r coopera t ion .

1.2 Elimination of the Mutual Couplings Between the Phases and the "Uniform Section" of a Ground C i r - cu i t - Ref. 13 shows t h a t a l l the mutual couplings

between the 'knnform sec t ion" (A,B) of a ground c i r c u i t and the phases of the c i r c u i t s 1 ,2 ,3 , . . . , can be rep laced by a cur ren t source (IR),cOnneCted t o t h e ground cir- c u i t a t the terminal po in t s (A) , (B) of the "uniform sac t ion" (See Fig. 1.2 ex t r ac t ed from Ref. [ 1 3 ) . This powerful transformation ca l l ed by the authors"Decoup1ing Technique" i s only a spec ia l app l i ca t ion of t h e Norton The or em.

38 MNCIT, IMPEDRNClS

31 TRRNSU, IKPEIlllNCES

61 C l R C U l l ninrtm

ANNEX I - THE "SELF-NEUTRALIZED CURRENT" CONCEPT

The " se l f -neu t r a l i zed ' ' c u r r e n t concept i s the "Decoupling Technique" which i s the S tep "Decoupled Method" [ 12. Method a r e presented below.

1.1

4

For e a s i e r r e f e r e n c e t h e b a s i c concepts

Charac t e r i za t ion of an "Uniform Sec t ion"

F ig . 1.1 shows an e x t e n t i o n of t he ground-wires of a l i n e between o i n t s (A) and (B). encompassing 30 l i n e spans . Ref. fl] s t a t e s t h a t t h e ground-wire sec- t i o n (A,B) can be Pa l l ed a "uniform sec t ion" p rov id ing the fo l lowing parameters can be assumed a s uniform: 0 (21) self-impedance of t h e ground-wires i n n/km 0 (zml),(zmZ), (zmg), ... mutual impedances between(A,B)

and l i n e s 1,2,3,,,, i n R/km (11) , ( I 2 ) , ( I ~ ) ~ , . zero sequence c u r r e n t s i n Amperes, f lowing through the phases of the l i n e s , 1 , 2,3,..,1n- f l u e n c i n g the "uniform sec t ion" . It should be remembered t h a t non-uniform spans and

d i f f e r e n t tower f o o t i n g r e s i s t a n c e do n o t c o n f l i c t w i t h t h e "uniform sec t ion" concept .

A "uniform sec t ion" can also be determined on an underground conductor such as: t h e shea ths of a ca- b l e ; an in t e rconnec t ing copper conductor bu r i ed i n the s o i l ; a p i p e l i n e , e t c . These conductors a r e r e a l l y lad- d e r c i r c u i t s w i th d i s t r i b u t e d parameter , which can be r ep resen ted , w i th any d e s i r e d p r e c i s i o n , by a lumped pa- ramater l adde r c i r c u i t w i th enough number of p i s (See F ig . 4 ) .

3 LIHES 90 HUIUPL

COUPLlHCS 30 L O H C I l U D l N R L

1MPEDRNCES 31 TRRNSUERSr

IHPLDllHCfS 154 C I R C U I I

ELEnEMS

--

38 PIS fl

Fig. 1-2 - Elimination of the Mutual Couplings With Phases of

of t h a t

The ca l cu la t ion of (1,)is c a r r i e d ou t as fo l lows:

(1) ground

The parameter (U) i s genera l ly used t o r ep lace the r e l a t i o n (zm1/z1) on the c i r c u i t .

The cur ren t source (I ) produce on the c i r c y i t the same e f f e c t s t h a t would f e caused by the l i n e c i r - c u i t s and the mutual couplings, which need no longer t o be represented.

I. 3 The Concept of "Self-Neutralized" Current i m -

pedance (21) produces a s vo l t age drop ( I R . z ~ ) , i n V/Km, which ba lances exac t ly the induced vo l t age (v) , i n V/Km. For t h i s reason ( IR) was c a l l e d by us "se l f -neut ra l ized" cu r ren t by the au tho t s :

The flow of (Ip) along the "uniform sect<on"

I R - -v1 /21 a ' . I R Z l " V I

Hence the flow of (IR) along tho uniform s e c t i o n (A.B) do n o t cause any n e t p o t e n t i a l clinnge along tllr ground c i r c u i t ,

1.4 Phys ica l Meaning of the "Self-Neutralized" Current

F ig . 1-1 and Fig . 1-2 above show t h a t the e f - f e c t s produced upon the ground c i r c u i t by the appl ica- t i o n of an uniform vo l t age (VI) along the "uniform sec- t ion", can be rep laced by a cu r ren t source absorbing the "se l f -neut ra l ized" cu r ren t (IR) from (A) and i n j e c t i n g i t back upstream a t po in t ( B ) .

This means t h a t the app l i ca t ion of an uniform vol tage along a ground c i r c u i t between po in t s (A)and(B), corresponds t o the absorb t ion of a c e r t a i n amount of cuurren t ( IR) from (A) and the i n j e c t i o n of ( IR) upstream, a t the po in t (B). This corresponds t o R cur ren t t r a n s f e r from (A) t o (B). Fig.1-1- Charac te r iza t ion of a "Uniform Section"

161 1

ANNEX I1 - SOLUTION OF LUMPED PARAMETER LADDER CIRCUITS

I n t h e nex t s e c t i o n s the technique of lumped parameter l adde r c i r c u i t manipula t ion i s der ived , s t a r t i n g w i t h I n f i n i t e Ladder C i r c u i t s (ILC). For t h e sake of c l a r i t y and s i m p l i c i t y , a l l t he c i r c u i t s pre- s en ted i n t h i s Annex I1 show ladder c i r c u i t s composed by r e s i s t i v e elements only. However, a l l t h e techniques i n these papers can be used t o so lve l adde r c i r c u i t s composed by complex impedances. The u t i l i z a t i o n of com- p l ex c u r r e n t sources e x c i t i n g the c i r c u i t s i s a l s o al- lowed.

11.1 SOLUTION OF AN INFINITE LADDER CIRCUIT (ILC)

Although (ILC) is a f i c t i c i o u s c i r c u i t , i t s care- f u l study is valuable i n order t o prepare the s o l u t i o n of the (FLC)'s which a r e very o f t en found i n grounding s tud ie s ( subs ta t ions and l ines) ,aswel l as i n the s t u d i e s of t ransmiss ion l i n e s inf luenc ing p ipe l ines o r telephone l i n e s . 11.1.1TerminalImpedancesof an I n f i n i t e Ladder C i r c u i t

Fig. 11.1 shows a ladder c i r c u i t assumed wi th an i n f i n i t e number of p i s . Each p i i s formed by lon- g i t u d i n a l impedance (2,) and the t r ansve r se impedance (ZT), both i n Q.

t h e

I H s:ima,ik

F i g . I I . l - I n f i n i t e Ladder C i r c u i t (ILC)

The te rmina l impedance (2,) seen t o the r i g h t s i d e from the node (n) i s defined as the s ing le impedance ab le t o rep lace a l l the c i r c u i t connected t o the node(n) t o the r i g h t s i d e . Looking t o t h e r i g h t s i d e from the node (1) o r from the node (2) w e can see i n cases an i n f i n i t e number of p i s . Therefore, looking t o the r i g h t s i d e from the node (1) o r from the node (2)we must s ee the same t e rmina l impedance (2,). Therefore,we can rep lace by (zE2! a l l the c i r c u i t seen t o the r i g h t s i d e from (2) and wr i t e the following expression f o r the t e r - minal impedance ( 2 ~ 1 ) seen t o the r i g h t s i d e from the node (1) (See F ig . 11.2).

'E1 + ('T / 1 'E21 'E1 I '32 ' E

both

% - ZL + (2, I I ZE)

ZE - ZL + (2, * zE)/(zT 'E)

Solving i n terms of (2,) i t comes (from [ 17) : (I) ZE = ZL12 + 3 2 /4 + ZL . ZT i n Q

For example assuming an (ILC) wi th Z = 2Q,ZT = 1.50 i t comes: ZE=3 Q L

r - -

~~ ~~

Fig.II.2-Terminal Impedance Seen From the Node (1) and ( 2 ) of an (ILC)

fI.1.2Voltage and Current a t Any Node of an I n f i n i t e Ladder C i r c u i t

The formulae introduced below were der ived us ing the concept of terminal impedance (ZE), as follows: a ) L e t us assume a cu r ren t source appl ied a t the lefmost node of an (ILC),see Fig.II .3.The cu r ren t a r r iv - i ng t o each node sees the impedances (ZT) i n p a r a l l e l with the impedance (which r ep laces a l l the c i r c u i t t o the r i g h t side).$or ins tance . the cur ren t I = 3.333 Amperes, flowing along the c i r c u i t s e c t i o n (1-2),reaches the node (2) and sees (ZT = 1.5 Q) i n p a r a l l e l w i th (23' 3 Q) (which rep laces a l l t he c i r c u i t seen from ( 2 ) t o t h e r i g h t s ide . Therefore, the cu r ren t (1 l ) reaching the node (2) w i l l be subdivided i n t o two components (12) and (ip) flowing through (ZE) and (%) r e spec t ive ly :

12 ( i n Amps) flows rightwards through (2-3). i2 ( i n Amps)

The cu r ren t s flowing a long the c i r c u i t shown i n Fig. 11. 3 below, were computed using mat r ix methods i n order t o allow a d i r e c t v e r i f i c a t i o n of the formulas presented i n t h i s paper.

flows t o ground through (2,).

F i g . I I . 3 Current D i s t r ibu t ion Along an (ILC)

b) A s (2,) and (2 ) seen from each node of the (ILC) a r e the same,, the "Ecurrent d i s t r i b u t i o n factor"($), computed us ing ( 2 ) and (ZE),applies f o r a l l the (1LC)nodes ( see Fig. 1 1 . 4 7 :

K = ZT/ (ZT + ZE) (2)

For example, using the Fig. 11.3, i t comes:

K = 0.3333 I2 = (K) I1 = (0.3333)3.3333 = 1.1111 (Amperes)

i2 = (l-K)I1- (1-0.3333)3.3333 - 2.2222 (Amperes)

r' 2 E

I VI1-K I

. Fig.II .4-"Current D i s t r ibu t ion Factor"(K) f o r and (ILC)

c) These procedures can be genera l ized a l lawing the d i r e c t ca l cu la t ion o f : vo l t age t o ground (Vn);current t o ground ( i n ) ; cu r ren t r ightwards (In), a t a genera l node (n) of an I n f i n i t e Ladder C i r c u i t exc i t ed by a cu r ren t source ( I ) appl ied t o the l e f t t e rmina l node.

In - 1.K" i n Amperes

in = I.(I-K)P-' in Amperes

Vn = i,.ZT i n Volts

1612

For example (see Fig. 3 ) :

I3 = 10(0.3333)3 - 0.3704

1 3 - lO(1-0,3333) (0.333312 - 0.7404 Amperes

Amperes

V3 0.7404 (1.5) 1.11 Vol ts 11.~3Voltage Decay Along the I n f i n i t e Ladder C i r c u i t a ) Let us msume the: i n f i n i t e ladder c i r c u i t wi th nodes 1,2,... shown i n Pig. 11.3.

The c u r r e n t source (I) appl ied a t the node (1) produces a vo l tage (V,) a t the node (1) such t h a t :

b) which the vol tage t o ground (V (a.Vl) of t h e i n i t i a l va lue (Vyf. Vnl = (In1)Z, =

aV1 - a(I1)% - a(IK1)ZE V n l - aVl-rr(Knl)%- a(I.K)ZE+Knl - eK+Kn1-l - a

(nl-1) Iln(K) = h ( a ) n l - 1 4 Iln (a)/Rn(K) c)

(a = e-' = 0.368) (a s i m i l a r reduct ion f a c t o r i s con- s idered by circui t theory t o der tenoine the '!tima con- s tan t ' ' ) .

I f w e rep lace t h i s va lue of (a) i n the express ion (6). w e have:

n l = 1 4

I n t h i s case (n,) is t h e node number of n i t e ladder c i r c u i t a t which the vol tage f a l l s t o of the vol tage a t the node (1) where t h e c u r r e n t was appl ied (See Fig . I I .5 ) .

v i = (1) zE* It i s p o s s i b l e t o determine t h e node number ( n l ) a t

) f a l l s t o a percentage

( 6 ) The reduct ion f a c t o r (a) can b e chosen as:

Rn (e-1) -1 Iln(K)--1+ m t h e i n f i -

36.8% source

Fig. 11.5-Current Decay Along an (ILC) and Ladder's Space Constant" (SC) ( o r C h a r a c t e r i s t i c Length)

d) Considering (S) as the e x t e n t i o n ( i n km) of each p i composing the ladder c i r c u i t , i t i s p o s s i b l e t o determine the d i s t a n c e from the node ( 1 ) a t which the v o l t - age f a l l s t o 36.8% of the p o t e n t i a l (VI) a t che node(1). This d i s t a n c e i s c a l l e d "Space Constant" (SC) 0r"Charac- ter is te Length" (CL). SC = ( S ) ( n l - l ) = S - = - - s

( !Li iK)) (In(K) in km

( 7 ) -S sc = - Rn(K1) in km

K1 = Module of the complex (K) "Charac te r i s t ic Length" (OK "Space Constant")is no t

a new concept 77. However t h i s paper in t roduces a new formula which permits the determination of t h i s pa- ramater using a lumped parameter approach, i n s t e a d of the c l a s s i c a l d i s t r i b u t e d parameter approach .

11.2 SOLUTION OF A FINITE LADDER C I R C U I T WITHOUT SPECIAL TERMINATIONS

It i s shown below t h a t t h e v o l t a g e and t h e c u r r e n t t o ground a t any node (n) of a (FLC) wi th (N) nodes,((N) ranging from 2 t o (-), e x c i t e d by a c u r r e n t source (I), appl ied t o t h e l e f t t e rmina l ) can be computed by simple exac t formulae.

It i s shown f i r s t i n i t e m (11.2.1) below t h a t the c u r r e n t t o ground i n a (FLC) can be represented by a i n f i n i t e power series. I t e m (11.2.2) below def ines t h e genera l express ion of t h i s series. I t e m (11.2.3) shows a n exac t formula (express ion (11) of Table 11.1) which computes the exac t a d d i t i o n of t h e terms of the i n f i - n i t e series, allowing a very simple manipulation o f (FLC)'s. Numerical examples of the u t i l i z a t i o n of t h i s formula a r e shown i n i t e m (11-2.6) below.

II.2.1Voltage and Current t o Ground a t a llode(n) of a

a) The Fig.II.6Ashows a f i n i t e Ladder c i r c u i t v i t h 5 nodes (N - 5 ) . e x c i t e d by a c u r r e n t source (I), appl ied t o the node (0). This c i r c u i t was so lved by mat r ix methods t o allow an easy v e r i f i c a t i o n of the c a l c u l a t i o n procedures introduced below. The s o l u t i o n of t h i s (PLC)

(FLC) Defined as a Power Series

Fig. 11.6A-CFLC) Solved by C l a s s i c a l Methods

can be obtained a l s o by computing the parameters (2,) and (K), the "current d i s t r i b u t i o n f a c t o r " a t the nodes (1, 2, 3. 4 , 5 ) , assuming t h e ladder c i r c u i t has an in- f i n i t e number of p i s . The s o l u t i o n procedure i n shown i n P i g . 6B.Should t h e ladder be i n f i n i t e , t h e cur ren t reach- Lng the node ( S ) , flowing a long the c i r c u i t s e c t i o n (4-5) would be (IK4), ( see express ion ( 3 ) - (above). The curren t f lowing through the next s e c t i o n (5-6) would be (IK5). However, the remainder of the (ILC) seen from (5) t o the r i g h t s i d e is f i c t i c i o u s . I f w e i n t e r r u p t the in- f i n i t e ladder c i r c u i t j u s t a t the r i g h t of the node (5) and i n j e c t the c u r r e n t (IK5) t o the l e f t s i d e , w e s a t - i s f y simultaneously the 2 phys ica l c o n s t r a i n t s :

- The c i r c u i t s must be i n t e r r u p t e d a t the node (5) - The c u r r e n t must be zero a t t h e node (5) b) Af te r t h i s , we have t o compute the c u r r e n t d i s t r i - bu t ion along the l a d d e r c i r c u i t caused by the i n j e c t i o n of (IK5) i n t o t h e the node ( 5 ) . This can be accomplished assuming an (ILC) extended t o t h e l e f t s i d e and e x c i t e d by a cur ren t source (IK5) appl ied t o the rightmost terminal node. Therefore, i t i s a s i f the c u r r e n t (IK5)were " re f lec ted" a t the open node (5) and f lew backwards through the "auxi l ia ry" nodes ( 6 , 7 , 8 , 9 , 10) . which corresponds t o the r e a l nodes (5, 4 , 3 , 2 , 1). c) The second a u x i l i a r y (1LC)extended t o the l e f t s i d e must be opened aga in j u s t a t the l e f t of the node ( l o ) , because the remainder of the (ILC), seen from ( l 0 ) t o the l e f t s i d e , i s f i c t i c i o u s . The c u r r e n t (1K")is " i n j e c t e d t o the r i g h t s i d e " flowing along the "auxi l ia ry" nodes (11, 1 2 , 13, 1 4 , 15) which correspond t o the r e a l nodes (1, 2. 3, 4 , 5 ) . A t the node (15) the f i r s t " r e f l e c t i o n cycle" i s completed. The c u r r e n t (IK15)i.s r e f l e c t e d back and a new " r e f l e c t i o n cycle" begins . The process con- continues e n d l e s s l y and the magnitude of the c u r r e n t s reduce each cyc le .

1613

p 18 9 1 0 I 7 6 I Oh I 'lSl7 I I

I

12 I 13 I 14 15 , !p 11 I I

5:. OPM

Fig.II.6B- Current to Ground in a (FLC) - Formation

d) Let us assume now that the calculation of (i3), flowing from node (3) to ground, is required. This current (i3) will be the supperposition of the currentcl flowing to ground from the nodes ( 3 , 8, 13, 18, 23, ...

The value of each one of these currents can be com- the terms

of the Power Series

e t c ) ,

puted using the expression (3) above. Adding we obtain the following expression for (i3):

i3 = I(1-K)(K2 + K7 + K1' + K17 + K22 + ....) Assuming I = 10 1 0 A ZL 2R ZT = 50R it comes:

i3 = 1.9225 Amperes (compare with Fig. II6A)

The method describe above is related to the Compensation Theorem and its conditions are match in derivation.

The correction of the results can be easily verified by the numerical example shown in item (11.4.a) in which the process is successfully applied even for the case of a single pi (minimum(FLC)configuration).

It can also be verified that any cuirent to ground shown in Fig. 6A, computed by matrix inversion, can be exactly computed by a power series.

In the next items it is shown that the addition of the terms of this series can be obtained exactly by a simple formula. This means that the exact computation of current and voltage at any node of a ladder circuit.with any size,can be obtained by a simple formula. avoiding the corresponding storage area requirements for matri- cia1 representation of the circuit.

11.2.2Porrer Series Representing the Current to Ground (General Expression) It can be observed that the exponents of (K) used

in the above power series can be expressed in a general way, in terms of the node (n) and of the total number of nodes (N) of the (FLC). The general power series expression which allows the computation of (in) is shown in Table 1, item A. IL2.3 Exact Addition of All the Terms of the Power Series

The items (B) and (C) of Table 1 shows that the exact sun, of all the terms of the power series can be computed directly by means of a general formula shown in item (D) of Table II.1.This formulaallow the direct calculation of the current to ground (in) and the voltage to ground (V,) at the node (n) of a (FLC) with (N) nodes. The (FLC) is excited by a current source (I) ,connected to one of the (FLC) terminals and without special terminations (i.e the transverse impedances to ground connected to the right and left terminals are also equal to (ZT), as in the intermediate nodes).

A) General Expression of the Power Series

a I g - ~7 Kn-l + ~(zN)l+(n-1)+,(2N)2+(n-l) n

E( 2N) 1-n+K(2N) 2-n+K(2N) 3-n + K(W3 + (n-l)+.,. +

(

( 8 ) + . . . .a} 8) Simplification of the Power Serie

C) Addition of All the Terms of the Power Series

AS c < 1 it comes: c1+c2+c3+. , . - - C = - KZN 1-C 1-K2N (10)

D) Final Formulae

A=- KZN 1 -K2

Vn - inZT in Volts (12)

Table 11.1-Solution of a Finite Ladder Circuit Without Special Terminations - Derivation of the Final Formulae

n-1 VI: 14.285721 nr2 ~ 2 : 5,7142781

Fig.II.7A-Solution of a (FLC) With Only 2 Nodes

n z l ul-122,681 n 3 v5-85.7BU

Fig.1I.B-Solution of a (FLC) With 5 Nodes

1614

II.2.4Numerical Examples

a) (ZT - 23). The cu r ren t ( 1 - lo"/O )Amperes i n t o the node (1 ) . Using the e q i e s s i o n s (11). and of Table r ead i ly c o m p u t d 2, = 9 + ,/y + ZL . zT = 4 . w "

K = - zT - 0.3138593

The Fig. 1 1 . 7 A shows a ( F W with (NE 2) ; !ZL - 3 . 3 ) , is i n j e c t e d

(12)

z ~ + z ~ node 1 + v1 = 14.28572 V o l t s , i l = 7.1428 Amperes

node 2 * v2 =

The co r rec tness of t hese r e s u l t s can be e a s i l y v e r i f i e d by solving the c i r c u i t using conventional methods.

5.714278 Vol t s , i 2 = 2.857139 Amperes

b) The Fig.II7B shows a(FLC) with (NI 5); (ZL = 2 3 ), (ZT~ - 50 Q),

The cur ran t (I - 10.0 /&) Amperer i e i n j e c t e d in- t o the node (1) . Ueing agafn the expreeaion (11).(12) of Table I I . lwe can compute d i r e c t l y the vol tage and cur- r e n t a t any node. For examples:

zE = i i .049a76

n = 1-+ v1 = 122.08V

n = 5+ v5 - 85.70V

K = 0.8190024

il = 2.4466A

i5 - 1.714QA

- f -;$- -;a - - t I I

Fig . 11-8 - Replacement of a Ladder C i r c u i t by an Equivalent P i

11.3 REPLACEMENT OF A DECOWLED LUMPED PARAMETER LAD- DER CIRCUIT BY AN EQUIVALENT PI

A gene ra l method i s descr ibed below which can be app l i ed t o uniform lumped parameter ladder c i r - c u i t s w i th any number of p i s (from 1 t o i n f i n i t e ) .

11.3.1 GENERAL METHOD

a ) Assume t h a t a l l t h e p i s between (A) and (B) ( i n - c lud ing t h e c i r c u i t elements connecting (A) and(B) t o ground)were disconnected from t he o r i g i n a l c i r c u i t (See Fig. 11 .8 ) ;

b) Apply a c u r r e n t I = 1 . 0 / 2 Amperes i n t o t h e node (A)and compute the r e s u l t i n g voltages(VA)and (Vg) , i n Vo l t s , a t t h e nodes (A) and (B). The c a l c u l a t i o n of (VA) and (VB) can be made by simple formulae as shown i n subsec t ion 11.3.2 below;

c ) The equ iva len t p i is formed by t h e span impedance (Q) and by t h e two connections t o ground (P ) . These parameteres can be determined using t h e fo l lowing expres s ions :

P = 'VA + VB)/1 i n Q

d) The expres s ions above came from the s o l u t i o n of t h e fou r s imultaneous equat ions below. The c u r r e n t s ( I p ) and ( I ) are included i n the equ iva len t p i shown i n the F i g . T I . 8 .

11.3.2 CALCULATION OF (VA) AND (Vg)

The po in t s of t h e segregated c i r c u i t shown i n Fig. 11.8 are numbered from (1) t o (N). The uniform p i s are formed by t h e impedance (ZG), (2 , ) .The fol lowing formulae can be used:

(3) K 2N

A = 1 - K2N

V = I.ZG(l-K) (1+A) + A 1 Volts ( 4 ) Kn

Zs 5 uniform l o n g i t u d i n a l impedances, i n Cl ZG = u n i f o r m t r a n s v e r s e impedance, i n $2

The expression (4) al lows t h e d i r e c t c a l c u l a t i o n of t he p o t e n t i a l t o ground (VI, a t any p o i n t (n) between (1 ) and (N) i n c l u s i v e . Therefore l e t t i n g n= l we o b t a i n (VA) and l e t t i n g n = N we o b t a i n (VB).

The d e r i v a t i o n of t he above formulae are shown i n i t e m 11.2 of Annex 11.

11.3.3 REMARKS ABOUT THE RESULTING EQUIVALENT P I Considering (S) as t h e e x t e n t i o n ( i n km) of each

p i forming the ladder c i r c u i t , i t i s poss ib l e t o determine the d i s t a n c e from the node (1) at which the vo l t age f a l l s t o 36.84 of t h e p o t e n t i a l (Vi) a t t h e node ( 1 ) . This d i s t a n c e i s c a l l e d "Space Constant"(SC) , It i s a l s o c a l l e d C h a r a c t e r i s t i c Lenght (CL)

K1 = Module of t h e complex (K)

- s i n km (5) sc = - En (K1)

The d e r i v a t i o n of t he expres s ion (5) above i s shown i n i t e m 11.1.3 of Annex 11.

If the ex tens ion of t h e l adde r c i r c u i t i n t e r - connect ing 2 "monitored po in t s " i s equal OK l a r g e r t han 3(SC) i n k i lome te r , t h e expression(l),(2),(3),(4)shows t h a t t he va lue of t h e impedance (Q) of the equ iva len t p i approaches i n f i n i t e while t he va lue of (P)approaches

This means t h a t i f twol'monitored po in t s " a r e interconnected by a l adde r c i r c u i t l a r g e r than 3 (SC), then the ground c i r c u i t is opened between these two po in t s .

(zE> *

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Dimensioning of nearby substations interconnected ground system