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Method for reducing losses in distribution feeders A Kuppurajulu, S Elangovan and T Krishnaparandhama Department of Electrical Engineering, Indian Institute of Technology, Madras-600 036, India P Subramaniam Department of Electrical Engineering, Sri Venkatesware University College of Engineering, Tirupati, India A method for reducing annual energy losses in a power system is presented. The technique deals with the instal- lation of shunt capacitors and the changing of conductor size in order to reduce losses. An equation is derived for the net annual saving, and analytical expressions are obtained for the capacitor size and the conductor cross-section that yieM a maximum net annual saving. A simple computer program is developed for computing the numerical values. The technique is applied to a set ofll kV feeders ofan existing system, and the results obtained indicate that the method is an efficient one for reducing losses in a power system. I. Introduction Economy of generation, transmission and distribution of electrical energy is absolutely essential in present-day power systems. The capital expenditure on the required rate of power generation and transmission is so large that even a marginal increase in the efficiency of operation by mini- rnizing losses is economically significant. Literature available on minimization of losses deals mainly with the application of shunt capacitors. Cook 1 gives a method of reducing losses by shunt-capacitor application. Chang2 describes a method of applying shunt capacitors for voltage control and loss reduction. Tseng 3 presents a method of obtaining the size and location of capacitors for maximum power loss reduction. For a given operating condition, the transmission losses are also closely related to the conductor size. The proposed method gives the additional conductor cross- sectional area required and the size of the shunt capacitors to be installed for reducing annual energy losses with a maximum net annual saving. The investigation deals with primary feeders consisting of a number of line segments. A line segment is a part of the feeder between nodes of electrical importance where a change in conductor s~e occurs or where voltage-regulating equipment or large individual loads are located. The loca- tions of the distribution transformers along the feeder are considered to be the points at which loads are concentrated as this is a realistic and accurate representation. One prob- Received: 5 November 1980, Revised: 5 June 1981 lem generally associated with minimization of losses by shunt-capacitor application is the availability of clear poles for such installations 4 The present method overcomes this problem by locating the capacitors at the distribution transformers. The change in conductor cross-sectional area and the size of the capacitors to be installed are obtained in order to maintain the power factor close to unity, so mini- mizing current and losses. II. Proposed technique Figure 1 shows a distribution feeder with concentrated loads representing the distribution transformers scattered along the feeder. Shunt capacitors C1, 6"2 ...... Cn installed at load points are also shown in the figure. Let 1, = &, - jI.~ 12 =Ia2 --jim In = Ian -- jlrn The total load current is given by l= la -- jI r where &=Y'.&i i=1 and n #: Y. #i i=I The total capacitor current is i=l (1) (2) (3) (4) (S) Vol 3 No 4 October 1981 0142-0615/81/040193-04 $02.00 © 1981 IPC BusinessPress 193

Method for reducing losses in distribution feeders

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Method for reducing losses in distribution feeders A Kuppurajulu, S Elangovan and T Krishnaparandhama Department of Electrical Engineering, Indian Institute of Technology, Madras-600 036, India

P Subramaniam Department of Electrical Engineering, Sri Venkatesware University College of Engineering, Tirupati, India

A method for reducing annual energy losses in a power system is presented. The technique deals with the instal- lation o f shunt capacitors and the changing of conductor size in order to reduce losses. An equation is derived for the net annual saving, and analytical expressions are obtained for the capacitor size and the conductor cross-section that yieM a maximum net annual saving. A simple computer program is developed for computing the numerical values. The technique is applied to a set o f l l kV feeders o fan existing system, and the results obtained indicate that the method is an efficient one for reducing losses in a power system.

I. Introduction Economy of generation, transmission and distribution of electrical energy is absolutely essential in present-day power systems. The capital expenditure on the required rate of power generation and transmission is so large that even a marginal increase in the efficiency of operation by mini- rnizing losses is economically significant. Literature available on minimization of losses deals mainly with the application of shunt capacitors. Cook 1 gives a method of reducing losses by shunt-capacitor application. Chang 2 describes a method of applying shunt capacitors for voltage control and loss reduction. Tseng 3 presents a method of obtaining the size and location of capacitors for maximum power loss reduction. For a given operating condition, the transmission losses are also closely related to the conductor size. The proposed method gives the additional conductor cross- sectional area required and the size of the shunt capacitors to be installed for reducing annual energy losses with a maximum net annual saving.

The investigation deals with primary feeders consisting of a number of line segments. A line segment is a part of the feeder between nodes of electrical importance where a change in conductor s~e occurs or where voltage-regulating equipment or large individual loads are located. The loca- tions of the distribution transformers along the feeder are considered to be the points at which loads are concentrated as this is a realistic and accurate representation. One prob-

Received: 5 November 1980, Revised: 5 June 1981

lem generally associated with minimization of losses by shunt-capacitor application is the availability of clear poles for such installations 4 The present method overcomes this problem by locating the capacitors at the distribution transformers. The change in conductor cross-sectional area and the size of the capacitors to be installed are obtained in order to maintain the power factor close to unity, so mini- mizing current and losses.

II. Proposed technique Figure 1 shows a distribution feeder with concentrated loads representing the distribution transformers scattered along the feeder. Shunt capacitors C1, 6"2 . . . . . . Cn installed at load points are also shown in the figure. Let

1, = &, - jI.~

12 =Ia2 - - j i m

In = Ian -- jlrn

The total load current is given by

l= la - - j I r

where

&=Y'.&i i = 1

and

n

#: Y. #i i=I

The total capacitor current is

i=l

(1)

(2)

(3)

(4)

(S)

Vol 3 No 4 October 1 9 8 1 0142-0615/81/040193-04 $02.00 © 1981 IPC Business Press 193

L -

r~

Figure 1

l I 12 -

Distribution feeder with shunt capacitors

The existing feeder loss/phase is given by

LE (I2a "}" 12) pll + [([a [a 1) 2 + (I, / r l ) 2] p(12 11) A1 A2

1 2 n--I ) 2 ] P ( l n - - l n - 1 )

i=1 i=1 (6)

where P is the specific resistance of the conductor material, I i in general is the length up to the ith load point and A i is the ith segment conductor cross-sectional area. With an increase in conductor cross-sectional area and capacitor installation, the reduced value of the feeder loss/phase

Pll LR = [I2a + (It - Ic) 2] - -

(A1 +al)

p(t~ - tl) + [ ( I . - I ~ , ) 2 + ( G - I~ - . r . _ I ~ , ) q - - + . . .

( A 2 + a s )

+ [(Ia -- i=IZ Iai) + --Ic-- i~__l I ' i+i=l Ici

p ( t . - l . _ l ) x (7)

(.4. + a.)

where a i in general is the additional conductor cross- sectional area to be provided for the ith section. Saving in losses/phase is

L s = LE - LR (8)

Allowing for interest and depreciation on the capital needed to increase the conductor cross-sectional area and install capacitors, the net annual saving due to reduced losses is

S = KI(LE - L R ) - - Ks Ic - K3 [al/1 +a2 (12 - /1) + . - .

+an(ln - I n - 1 ) ] (9)

where Kx is a constant to convert saving in losses into annual cost of energy saved, Ks is the annual cost of capacitor installation per amp&e of current drawn by it, and K3 is the annual charge on capital needed per unit increase in conductor volume.

The net annual saving S is a function o f l c t , Ics , . . . , Icn and al, as, . . . , an..For the saving to be maximal, the first partial derivatives of S with respect to the Ics and as must be zero, and the second partial derivatives must be negative. Differentiation of equation (9) partially with respect to the IcS yields

OS pl~ = 2 K 1 ( A ~ - - OI~, + a,) (I, Ic) - Ks = 0

ll (12 II ) OS (At +al) (A2 +a: ) 31cs - 2 K l p - - ( l , - le ) + - -

× ( I r - - l c - - l , l + I c x ) J - -K2 =0

OS

31,,,

ll - 2 K , P [ ( A , + a , ) ( I , - l c ) + ( l s - _ l l )

(A= + a=)

x (I, - Ic - 1,1 +1cl) + ' ' " + (ln -- ln--l) (.4. + a.)

n_l ] x ( I , - I c - ~" ( I , i - I c i ) - K s =0

i=1 (10)

Differentiating equation (9) with respect to the as and equating to zero, we obtain

3S

cqal

~S

c3a2

Pll - K1 [Ia 2 + ( I , - I c ) 2] K3 la = 0

(A, +a,) 2

- K , [ (Ia - I a t ) 2 + ( I t - I c - Ia l + I c l ) 2] - - p ( l s - l l )

(A 2 +a2) 2

-- K3(/: - ll) = 0

- K1 - ~" lai + I , - I c - • ([ri - Ici) ~an i=1 i=1

p(l n - ln_l) x - K , ( I . - l n -1) = 0 (11)

(A. +a.) 2

The two sets of equations (10) and (11) give the solution for the Ics and as as

K:Ia I c = I r - (12)

( 4 K , K , p l , ~ - K ~ ) 1'2

2K~ p l~ a l - ( z , - g ) - A 1 ( 1 3 )

K2

IKlp~I/2(Ia -~11a i ) -A n H ~ 1 qn= \ -~3 7 i=1

(14)

lo , = Ic - I , + I , , (15 )

n--1 Icn=Ic- l r+ Z (Ir i-Ici)+lrn n > l ( 1 6 )

i=1

It can be observed from equations (10) and (11) that the second partial derivatives are negative. Hence the Ics and as obtained above give S a maximum value, which is the net annual saving that can be obtained by increasing the con- ductor cross-sectional area and installing shunt capacitors.

It appears from equation (12) that, in a borderline case, Ic may become negative. This equation can be written as

Ie = Ir - K ' Ia (17)

194 Electrical Power & Energy Systems

i.e.

I c = la( tan $ - K ' ) (18)

where K ' is a constant and 4~ is the power-factor angle, lc becomes negative if tan 4~ < K ' , that is, if q5 < tan-1 (K' ) . Thus I¢ becomes negative for a small value of ~, that is, under good power factor conditions, indicating that no capacitor installation is needed.

In practice, it is not possible to have conductors of different cross-sectional areas for different feeder segments. It is better to restrict oneself to two or three conductor sizes only. Further, it may be desirable to have capacitors installed at some of the distribution transformers only and not at all the transformers. To accommodate such practical con- straints, the general problem formulation described above can be suitably modified. For instance, if the same conduc- tor is to be used for ith and (i + 1)th segments of a feeder and no capacitor is to be installed at, say, j th and k th load points, then

(Ai+ 1 + a ' i + l ) = ( a i + a i ) (t9)

I c / = Ick = 0 (20)

With these changes in equation (7), solution for the Ics and as is obtained in the same way as described earlier in the section.

I I I . Example and discussion The proposed technique is applied to a set of existing 11 kV feeders operating under low-power-factor conditions to determine the size of capacitors to be installed and the increase in conductor cross-sectional area required for reduc- ing losses. The results obtained for a typical feeder are given in Table 1. The feeder operates at 0.8 power-factor lagging. Aluminium conductors are used. Values of various constants used in the example are:

p = 34.0 x 10-992 m K~ = 3.6 rupees/W power saved K2 = 100.0 rupees/A capacitor installation K3 = 5000.0 rupees/m 3 conductor.

The feeder has five segments as shown in Figure 2. The pro- posal is to have only two sizes of conductor and three shunt capacitors installed in order to minimize loss in the feeder. Under these constraints, a judicious choice is

L 2 3 3 . 3 8 A 25.57A j26.14A 2 O00mTOOOm 70 200m

14000m

Figure 2 Typical feeder

-:q 39.1zA

32.99A

zoo

39.12A

• to have one size of conductor for the first two segments and another size for the rest of the feeder length,

• to have the three shunt capacitors installed at the end of the first, third and fourth segments.

Making use of the general solution equations (12)-(16), the solution for the required configuration shown in Figure 2 is obtained as

I c = / r - - K2/~

(4K1 K3 P It z - K~) 1/2 (21)

2K1P ll a I -

K2 - - - - (1 r -- I c ) - - A I (22)

I c l = I c -- I r + ]r l (23)

(A2 +a2) = (A, +a, ) (24)

/ c 2 = 0 112

a3'= ~, K3 !

(25)

(Ia - Ia, - Ia2) - A3 (26)

(A 4 Jra4) = (A S + a s ) - (A 3 + a 3 ) (27)

2 lc3 = Ic - Ir + ~,, (Iri - Ici) + It3 (28)

i=1

3

Ic4 = Ic - I~ + ~ (Iri - Ici) + It4 (29) i=1

Ies = 0 (30)

Using the feeder data given in this section, equations (21)- (30) are solved and the results obtained are given in Table 1.

Table 1 Results obtained for the sample feeder

Existing cross-

Segment l, section, no. m m 2

Segment current, [II

Suggested cross- section, m 2

C, LE , LR, S, /1 F (kW/phase) (kW/phase) rupees

1 2 000 49 x 10 - 6

2 5 000 49 x 10 - 6

3 3 200 49 x 10 -6 4 4 000 37 X l 0 -6

5 5 000 37 x 10 -6

157.20 123.82 98.25 72.11 39.12

330x 10 -6 10.20 8.57 0.92 330x 10 -6 - 13.29 1.57 240 x 10 -6 7.85 5.36 0.79 240x 10 -6 5.70 4.77 0.61 2 4 0 x 10 -6 - 1.75 0.29

79460

Vol 3 No 4 October 1981 195

It can be observed from Table 1 that if the method is implemented, there is an appreciable net annual saving in addition to a saving in losses in the feeder. These results are obtained for an average feeder load of 50% of rated value. The savings would be much greater if the loading were 75% or more.

IV. Conclusion An approach is presented for reducing the annual energy losses in distribution feeders. A change in conductor cross- sectional area and the installation of shunt capacitors are proposed. Results obtained by this approach are promising and indicate that the method is an efficient one for reducing losses in power-system feeders.

V. 1

References Cook, R F 'Calculating loss reduction afforded by shunt capacitor application' IEEE Trans. Power Appar. & Syst. Vol 83 No 6 (December 1964) pp 1227-1230

Chang, N E 'Locating shunt capacitors on primary feeder~ for voltage control and loss reduction' IEEE Trans, PowerAppar. & Syst. Vol 88 No 5 (October 1969) pp 1574-1577

Tseng, M C 'Graphical method for determining the maximum loss reduction, the size and location of shunt capacitors on power distribution lines' presented at IEEE Summer Meeting (PES) Mexico (17-22 July 1977)

Cook, R F 'Optimizing the application of shunt capaci- tors for reactive volt-ampere control and loss reduction' IEEE Trans. Power Appar. & Syst. Vol 80 No 4 (August 1961) pp 430-444

Kuppurajulu, A and Raman Nayar, K 'Minimisation of reactive power installation in a power system' Proc. Inst. Electr. Eng. (May 1972) pp 557-563

Nanda, J and Bijwe, P R 'A novel approach for genera- tion of transmission loss formula coefficients', presented at IEEE Summer Meeting (PES) Mexico (17-22 July 1977)

Dev, P, Gaylard, B and Nicholson, J A 'Influence of conductor designs and operating temperature on the economics of overhead lines' Proc. Inst. Electr. Eng. (March/April 1971 ) pp 573-590

196 Electrical Power & Energy Systems