Electric Power Systems Research, 5 (1982) 191 - 198 191

T w i n C a p a c i t o r C o m p e n s a t i o n o f High V o l t a g e T r a n s m i s s i o n L ine

K. SRIKRISHNA and R. SRINIVASAN

Department of Electrical Engineering, Thiagarajar College of Engineering, Madurai 625 Ol 5 (India)

(Received January 25, 1982)

SUMMARY

Twin capacitor series compensation is high- ly effective in the reduction of the line imped- ance and thus improves the power transfer capability of the transmission system as a whole. An analytical approach is developed for the first time for the 'effectiveness factor'. A numerical comparison of performance factors such as transmission efficiency, shunt reactive voltampere requirements at the terminals and the line operating angle under a maximum received power criterion brings out the influence of compensation as well as loca- tion. It is shown that the scheme which places capacitors both at the sending and the receiv- ing end is preferable.

mended that satisfactory performance is obtained with a capacitor located at both ends.

2. LOCATION OF THE CAPACITORS

Consider the location of the capacitors to be at distance x km from both the sending and the receiving end, as shown in Fig. 1. The generalised circuit constants ABCD are in terms of hyperbolic functions, given below for the uncompensated line:

[ : : ] = [cosh(vl) Zesinh(vl ~

kYc sinh(vl) cosh(vl)

1. INTROD UC~ION

Series capacitors afford an excellent means of increasing the power capability of long lines to their economic maximum [1 - 4]. Ashok Kumar et al. [5] have calculated 'com- pensation efficiency' for various line lengths and locations with single, twin and multi- capacitor banks. Kimbark has rightly pointed out [ 5] that infinite sub-division of capacitors, though highly effective, is not practicable from location and cost considerations. This paper therefore limits its attention to two capacitor banks only. A symmetrical arrange- ment has been chosen so that in either direc- tion the maximum power flow will remain the same .

Analytic expressions derived for the 'effec- tiveness factor' and the curves presented bring out the relative influence of compensation and location. Subsequently, the maximum received power Prmax is compared for various locations. After comparison of the overall per- formance at different locations, it is recom-

sendmg end Xc Xc

II ~ . ~ x ~ ( l - -2~c) ~ ~ x

L KM ~4

Fig. 1. Location of capacitors.

With the twin capacitor arrangement the resultant line constants are given by the following:

[A0 B0] [ c o s h ( u x ) Z e sinh(vx)]

Co Do = Yc sinh(px) cosh(ux) ] ×

1 Ye sinh[v(l-- 2o01 cosh[v(l-- 2x)] J

lye sinh(vx) cosh(px) J

0378-7796/82/0000-0000/$02.75 © Elsevier Sequoia/Printed in The Netherlands

192

The matrix multiplication is performed and expressions obtained for constants B0 and Ao, as shown in the Appendix.

3. E F F E C T I V E N E S S F A C T O R

S o --

Zc sinh(v/) -- jXc{2cosh(ux) c o s h [ v ( / - x)] }

--X2yo cosh2(vx) s inh[v( / - - 2x)] (1)

= Zc sinh(vl) - -Xc{2j cosh(vx)cosh[v( / - -x ) ] }

--X2{y¢ cosh2(vx) s i n h [ v ( / - 2x)]} (2)

Neglecting resistance and assuming the line to be a lossless one for which the character- istic impedance is real and the propagation constant is imaginary,

Z ¢ = p + j q = p ( q = 0 ) (3)

v = a + j/] = j/] (a = O) (4)

B0 = j(p sin(/]/) -- X¢{2cos(~x) cos[/]( /-- x)] +

+ Xc Y¢ cos2(/]x) sin[/](/-- 2x)] }) (5)

IB01 = p sin(/]/) -- 2X¢{cos(px) cos[/]( /-- x)] +

+ 0.5X~Y¢ cos2(/]x) s in[ / ] ( / - -2x)]} (6)

= p sin(/]/) + 2Xc X effectiveness factor (7)

(1) The effectiveness factor is governed by the length of the line, location of the capac- itors, the magnitude of compensation and the inverse of the characteristic impedance.

(2) When the capacitors are placed at the terminals (x = 0) the effectiveness factor be- comes --[cos(/]/) + 0.5XcY¢ sin(/]/)].

(3) When both the capacitors are located at the centre (x = 0.5), the effectiveness factor becomes --cos2(/]//2) and is thus independent of the magnitude of X¢.

4. LINE D A T A

The following transmission line is consid- ered:

E s = E r = 3 3 0 k V

Line length = 600 km

r' = 0.0512 ~ / k m

x' = 0.512 ~ / k m

y' = 3.33 X 10 -6 ~ -1 km-1

For the above data the values of Zc and v are

Z¢ = 3 9 2 . 5 1 2 - - j 19.574 =p +jq

v = 0.065 14 × 10 -3 + j 1.306 × 10 -8 = ~ + j/]

5. C O M P A R I S O N O F E F F E C T I V E N E S S F A C T O R

The effectiveness factor is made up of two parts: the first part is independent of the mag-

.nitude of the compensation and the other part is a function of the compensation. The effectiveness factor is independent of the magnitude of the series compensation for location at the centre, and for all other loca- tions it becomes dependent on the magnitude of the series compensation. The variation of the second part of the factor with compensa- tion and with location is shown in Figs. 2 and 3, respectively.

The variation of the effectiveness factor with location and with compensation is shown in Fig. 4.

A perusal of the curves shows that an increase in the effectiveness factor is obtained as the location of the capacitors is shifted from the centre towards the ends with increasing compensation levels.

0,175

0.15

I o,~25 J

? ~ 0.05

/ ~ o c a b o n at / / terminals.

/

J ' J " I/3rd po,nt s.

75 50 75 100 175 150

X C ohms Fig. 2. Var ia t ion o f second par t o f e f fec t iveness fac to r wi th c o m p e n s a t i o n .

0425 ]

I 0 .I0 -I"~"

0 SO 100 150 200 250 300

dtstance from either end X KM

Fig. 3. Variation of second part of effectiveness factor with location.

0

- 0 . 5

-O,G •

-0.7.

-0.8

>_

M.O

dlstanc¢ fr'om either Qnd x K M

50 100 15(3 200 250 300 l l i t i

~X c = Z 5 ~ " ~ / X c = S 0 f t

X c =75~

Fig. 4. Variation of effectiveness factor with location of twin capacitors for various values of compensation.

6. PERFORMANCE ANALYSIS

It can easily be guessed that the effective- ness factor cannot be a good criterion for fixation of the compensation. For the proper selection of compensation, complete perfor- mance characteristics of the line are obtained by using the expressions given in the Appendix.

The constants Ao and B0 are required for calculating the power transfer. The expres- sions used are:

Ao = cosh(v/) -- j X c Y c sinh(v/) +

- - X 2 y ~ cosh(vx) sinh(vx) s inh[v( / - - 2x)]

193

B0 = Zc sinh(v/) +

- - jX¢(2 cosh(vx) cosh [v ( l - -x ) ]} +

- - X 2 y ¢ cosh2(vx) sinh[v(l -- 2x)]

For assumed alternative locations of the capacitors, the values of Ao and B0 are evaluated. Using the above values of IA01, IB0t, /~0 and a0, a computer program was developed for the calculation of P ~ , Ps, Pl and 77. The results are given in Tables 1 - 4 and Figs. 5 and 6.

6.1. Variation o f Prmax From Figs. 5 and 6 and Table 1 the follow-

ing conclusions can be drawn. (1) For each location, the power transfer

increases with Xc up to a certain value and thereafter decreases. Thus there is an 'opt imum' compensation for Prmax for each location.

(2) From Fig. 5 it can be seen that at low compensations P~m~x is practically indepen- dent of location. At high values of compensa- tion, maximum received power is obtained by locating the capacitors at a distance greater than 1/3 from the ends. However, Fig. 6, which gives a plot of maximum Prma~ with location, shows that the maximum value of P~m~ occurs when the location of the twin capacitor is at the centre.

(3) Table 2 provides a comparison of Prmax for different locations at the same total com- pensation. The twin capacitor compensation at the terminals gives a greater P~ma~ when compared to the compensation at either end alone.

In the previous section it was shown that the effectiveness factor increases with com- pensation level. However, as can be seen, such an increased compensation level leads to lower Prm~ and hence must to be discarded.

Table 3 shows the variation of transmission efficiency 77 with compensation for various power levels. It can be seen that transmission efficiencies with capacitors located at one- third points are marginally higher than those obtained with the capacitors located at the terminals. With the twin capacitor at the centre the transmission efficiency is very nearly the same as the transmission efficiency with the capacitors at one-third points: There- fore it would be preferable to locate the capacitors at the centre {single location) rather

194

1000

900

I 700

E

500

4 0 0 - -

J J

J

X c =12 5 S'L

J X c =100~

X c =75~

X c =50~-

Xc =25-n-

0 50 100 15,0 200 250 30"0

distance from ~,ither end X KM

Fig. 5. Variation of Prmax with location of twin capacitors for various values of compensation.

than have two capacitors, each being at a one- third point. However, the transmission effi- ciency with the twin capacitor at the termi- nals is greater than the efficiency with a single capacitor located at either end and hence the twin arrangement is preferable.

Table 4 presents a comprehensive compar- ison of the system with different capacitor locations for a total compensat ion of 200 ~ . As can easily be seen, the best performance is obtained with capacitors located at one-third points, closely followed by that obtained with a twin capacitor at the centre. However, from the point of view of operat ion, maintenance and control , location of capacitors at the terminals is bet ter than other locations. Such an arrangement does no t require the existing

lo0o

900.

I 800

7oo E

J

50 100 150 200 250 300 dis tance from either end ~c KM

Fig. 6. Variation of Prmax with location of twin capacitors for optimum compensation. NB. The ordinates give the highest values of power that can be transferred with a twin series capacitor scheme. The corresponding X c values are, of course, different.

line to be opened at the one-third points or at the centre. A brief comparison of perfor- mance with a capacitor at the receiving end only and twin capacitor compensat ion at both sending and receiving ends shows that, for a twin capacitor scheme,

(1) transmission efficiency is greater up to a power of 300 MW and is very nearly the same at higher loadings;

(2) the rating of a series capacitor is smaller up to a power level of about 300 MW;

(3) the total shunt reactive compensation requirement at the terminals is less up to a power level of 30P MW, slightly more at inter- mediate loads, and less at heavier loads;

(4) the line operating angle is smaller and Prm~x is greater.

Therefore, on the whole, it can be taken that the twin capacitor compensat ion is marginally superior. Further, this symmetrical scheme has an added benefi t in that the flow of power in either direction remains the same.

TABLE 1

Pr max (MW) against location and compensation: twin capacitor configuration

X¢ (~) Distance x from sending and receiving end (km)

0 100 200 250 300

25 401.18 407.17 410.88 411.67 411.79 50 462.35 477.01 485.84 487.25 486.78 75 548.67 576.74 593.37 594.77 592.16

100 673.09 721.97 752.16 752.15 744.22 125 823.74 879.91 937.61 941.12 932.64 130 840.02 879.48 944.81 955.71 959.52 140 794.10 725.39 799.01 855.25 912.42 150 493.46 220.85 251.62 380.60 574.40

TABLE 2

Comparison of Pr max (MW) for various compensations

195

Compensation Total compensation 2Xc ( ~ )

50 100 150 200 250

l S R 397.06 446.94 507.64

s & ' 1/2 ~ c I/2

S R 411.79 486.78 592.16 1/3 1/3 1/3

S ~ f ~c ....... R 410.88 485.84 593.37

578.61

643.28

673.09

744.22

752.16

644.35

790.45

823.74

932.64

937.61

TABLE 3

Comparison of transmission efficiency 77 (%) for various locations and compensations

Pr (MW) Total Single capacitor compensation (~) Centre Receiving end Sending end

Twin capacitor

One-third points Both terminals

100 50 96.46 96.47 96.24 96.61 96.57 200 50 94.45 94.39 94.14 94.50 94.38 300 50 91.43 91.24 90.92 91.44 91.17 400 50 85.43 83.55 -- 85.33 83.19

100 100 93.00 96.10 95.43 94.53 96.43 200 100 92.67 94.19 93.47 93.72 94.16 300 100 90.48 91.28 90.45 91.52 91.12 400 100 86.95 86.94 85.70 88.21 86.71

100 150 96.36 95.06 93.50 96.66 96.41 200 150 94.28 93.73 92.08 94.38 93.96 300 150 91.61 91.13 89.32 91.63 90.95 400 150 88.43 87.63 85.44 88.38 87.26 500 150 84.19 82.02 77.38 84.10 81.78

100 200 96.26 92.07 88.70 96.53 96.29 200 200 94.04 92.49 88.88 94.10 93.60 300 200 91.35 90.54 86.69 91.31 90.50 400 200 88.37 87.66 83.39 88.26 87.03 500 200 84.97 83.86 78.54 84.81 82.94 600 200 80.83 77.94 -- 80.67 77.39

100 250 95.86 84.55 77.42 96.59 95.86 200 250 93.26 89.26 81.13 93.36 92.62 300 250 90.25 88.90 80.21 90.01 89.08 400 250 87.43 86.96 77.63 86.55 85.30 500 250 83.61 84.12 73.71 82.90 81.22 600 250 79.88 80.44 67.31 78.97 76.62

7. CONCLUSIONS

The performance of the twin capacitor arrangement has been compared for various locations. The influence on the effectiveness

factor has been derived analytically. It is shown that, for the line considered, the scheme of twin capacitors at the terminals proves satisfactory both from the point of view of performance and of operation. The

196

T A B L E 4

P e r f o r m a n c e charac te r i s t i cs for a t o t a l c o m p e n s a t i o n o f 200

Pr Twin capacitor at one-third points (MW)

~7 Qr Qs ]Qr[ + [Qs] S 6 (%) (MVAR) (MVAR) (MVAR) (MVAR) (deg)

Single capacitor at centre

~7 Qr Qs IQr[ + ]Qs[ S 6 (%) (MVAR) (MVAR) (MVAR) (MVAR) (deg)

100 96.53 80.62 - -135.40 216.02 20.97 5.94 200 94.10 35.90 - -146.90 182.80 78.75 12.12 300 91.31 --22.98 - -146.22 169.20 182.10 18.66 400 88.26 --98.96 - -130.60 229.56 341.49 25.74 500 84.81 --198.18 --94.96 293.14 575.88 33.68 600 80.67 --333.24 --28.31 361.55 926.49 43.16

Twin capacitor at terminals

100 96.29 71.25 - -124.03 195.28 37.83 6.72 200 93.60 25.12 - -132.33 157.45 95.28 13.74 300 90.50 --37.44 --126.21 163.65 189.41 21.26 400 87.03 - -121.22 --101.55 222.77 363.81 29.58 500 82.94 - -136.16 --49.65 285.81 616.71 39.30 600 77.39 --411.28 54.84 466.12 1040.64 52.18

96.26 82.77 --135.54 218.31 17.46 6.08 94.04 38.78 - -145.58 184.36 72.15 12.38 91.35 --19.49 --143.13 162.62 169.98 19.06 88.37 --95.38 - -125.28 220.66 321.00 26.30 84.97 - -194.99 --86.69 281.68 543.63 34.45 80.83 - -331.89 --15.48 347.37 878.82 44.23

Single capacitor at receiving end

92.07 208.03 34.49 242.52 97.83 5.39 92.49 167.80 28.35 196.15 125.16 13.25 90.54 109.91 38.19 148.10 187.47 21.69 87.66 29.00 68.87 97.87 295.38 31.07 83.86 --87.10 131.40 218.50 473.07 42.27 77.94 --282.40 265.60 548.00 807.66 58.32

methods are sufficiently general and can be readily adopted to any line configuration. A final decision on the choice of compensation and location can be made from a comparison of the performance characteristics for all loca- tions and the relative cost of locating the banks at intermediary stations.

A C K N O W L E D G E M E N T

The authors thank Professor M. Maria Louis, Principal, Tb.iagarajar College of Engi- neering, Madurai-15, India, for providing the necessary facilities.

N O M E N C L A T U R E

A, B, C, D generalised circuit constants of line

A0, Bo, Co, Do resultant generalised circuit constants of line with series capacitor

E s sending-end voltage of line, kV

Er receiving-end voltage of line, kV

l length of line, km Pr receiving-end power, MW Prmax maximum power received,

MW r' resistance of line, ~2/km x' reactance of line, ~ / k m Y shunt admittance of line,

~-~-1

Z impedance of line, ~2 Zc characteristic impedance of

line, ~2

Greek symbols s0 argument of A0 /3o argument of B0 A0 argument of Do 8 phase angle between E s and

Er (line operating angle) propagation constant of line % transmission efficiency

R E F E R E N C E S

1 E. C. S tar r and R. D. Evans, Series capac i to rs for t r ansmiss ion circuits , Trans. A m . Inst. Electr. Eng., 61 ( 1 9 4 2 ) 963 - 973.

2 A. A. J o h n s o n , J. E. Barkle and D. J. Povejsil, F u n d a m e n t a l ef fects o f series capac i to r in h igh vol tage t r ansmiss ion lines, Trans. Am. Inst. Electr. Eng., Part 1, 70 ( 1 9 5 1 ) 526 - 536.

3 R. S. S e y m o u r and E. C. Starr , E c o n o m i c aspects of series capac i tors in h igh vol tage t r ansmiss ion , Trans. Am. Inst. Electr. Eng., Part 2, 70 ( 1 9 5 1 ) 1663 - 1670.

4 S. B. Crary and L. E. Saline, Loca t i on of series capac i to r s in h igh vol tage t r ansmiss ion sys tems, Trans. A m . Inst. Electr. Eng., Part 3, 70 ( 1 9 5 3 ) 1140 - 1151 .

5 B. S. A s h o k Kumar , K. Par thasa ra th i , F. S. Prabha- kara and H. P. Khincha , Effec t iveness of series capac i to rs in long d is tance t r ansmiss ion line, I E E E Trans., PAS-89 ( 1 9 7 0 ) 941 - 951.

A P P E N D I X

197

Constant Bo The resultant generalised circuit constants are given by the matrix

Co Do LYe sinh(vx) cosh(vx) ] 0 Y~ sinh[v(1 -- 2x)]

X [ ; - - j X c ' [ c o s h ( v x ) Z ~ s i n h ( v x )

1 [ Ye sinh(vx) cosh(vx)

After performing the matrix multiplication step by step,

B 0 =

3 Zc sinh[v(/-- 2x)][

/

cosh[v(1 -- 2x)] J

cosh(vx) cosh[v(/-- 2x)] Z¢ sinh(vx) --jXcY¢ cosh(vx) --sinh[v(l -- 2x)] Z c sinh(vx) +

+ Z 2 Y 2 sinh2(vx) sinh[v(/-- 2x)] --jX~ cosh2(vx) cosh[v(/-- 2x)] +

+ Z e sinh(vx) cosh(vx) cosh [v ( / - 2x)] - - jX e cosh2(vx) cosh[v(/-- 2x)] +

--jX¢ sinh(vx) cosh(vx) sinh[v(1 --2x)]

= Z¢ sinh(vx) cosh(vx) cosh[v(/-- 2x)] + Z¢ sinh2(vx) sinh[v(l -- 2x)] +

+ Z¢ sinh(vx) cosh(vx) cosh[v(/-- 2x)] + Zc cosh2(vx) sinh[v(/-- 2x)] +

-- 2jX¢ cosh(vx) sinh(vx) sinh[v(/-- 2x)] -- 2jX¢ cosh2(vx) cosh[v(/-- 2x)] +

- - X 2 Y¢ cosh2(vx) sinh [v(l -- 2x) ]

= Z¢ sinh(vx) cosh[v(l--x)] + Z¢ cosh(vx) s inh [v ( l - x)] +

-- jX¢{2cosh(vx) cosh [v(l -- x)] ) -- X 2 Yc cosh2(vx) sinh[v(l -- 2x)]

= Z¢ sinh(vl) -- jX¢(2cosh(vx) cosh [v(l -- x)] } -- X~ Y¢ cosh 2 (vx) sinh[v(l -- 2x)]

×

Constant Ao

A0 = cosh2(vx) cosh[v(/-- 2x)] - - j X c Y c cosh2(vx) sinh[v(/-- 2x)] +

+ sinh(vx) cosh(vx) sinh[v(/-- 2x)] --jXc Y~ cosh(vx) sinh(vx) cosh[v(/-- 2x)] +

+ ZcY¢ cosh(vx) sinh(vx) sinh[v(/-- 2x)] 2 2 --X~ Y¢ cosh(vx) sinh(vx) sinh[v(l -- 2x)] +

+ Z¢Yc sinh2(vx) cosh[v(l -- 2x)] - - j X ¢ Y e cosh(vx) sinh(vx) cosh[v(/-- 2x)] +

-- jZcY~ sinh2(vx)X¢Y¢ sinh[v(l-- 2x)]

= cosh2(vx) cosh[v(l -- 2x)] + sinh2(vx) cosh[v(/-- 2x)] - - j X ¢ Y e cosh2(vx) sinh[v(/-- 2x)] +

- - j X ¢Y ¢ cosh(vx) sinh(vx) cosh[v(l -- 2x)] + sinh(vx) cosh(vx) sinh[v(/-- 2x)] +

+ cosh(vx) sinh(vx) s i nh [v ( l - 2x)] --jXcY¢ cosh(vx) sinh(vx) cosh [v ( l - 2x)] +

- - j X ~ Y¢ sinh2(vx) sinh[v(l -- 2x)] --X~ y2 cosh(vx) sinh(vx) sinh[v(l -- 2x)]

= cosh(2vx) cosh[v(/-- 2x)] + sinh(2vx) sinh[v(/-- 2x)] --jXeYe cosh(vx) sinh[v(/--x)] +

- - j X ~ Y c sinh(vx) cosh[v(/--x)] 2 2 - - X ~ Y¢ cosh(vx) sinh(vx) s inh [v ( / - 2x)]

= cosh(v/)--jXcY¢ sinh(v/)-- 2 2 Xe Ye cosh(vx) sinh(vx) sinh[v(/-- 2x)]

198

Equations used

EsEr Pr--

IBol - - - c o s ( r i o - t ~ )

P r m a x -- - -

EsE~ IBol

E21Aol

IBol

P ~ = _ _ _

E21Aol - - cos(rio - % )

IBo[ (AI)

- - cos(rio - - So) ( A 2 )

EsEr

IBol c o s ( r i o + 6 ) + - -

E21Dol cos(rio - Ao)

IBol ( A 3 )

EsEr Q r -

IBol - - - sin(ri o - - 8)

- - - sin(rio + 8) + - - EsEr

Qs - IBol

PI = Ps- -Pr

E~IAol - - sin(rio - - So)

IBol ( h 4 )

E21no[ sin(rio - - A o )

tBol ( A 5 )

( A 6 )

S = r a t i n g o f c a p a c i t o r , M V A R

= 312X¢ = $1 + $2 ( A 7 )