Evaluation of line loss under load unbalance using the complex unbalance factor

T.-H. Chen

Indexing terms: Complex unbalance factor, Line loss, Unbalance, Unrransposed transmission line

Abstract: The paper introduces simple criteria to evaluate the line loss under load unbalance by using the complex unbalance factor. The defini- tions of voltage and current unbalance factors have been extended from the conventional real values to the corresponding values. The complex- valued unbalance factor is composed of a magni- tude and an argument portion. The magnitude portion is commonly used to evaluate the effects of load unbalance. In contrast, the argument portion is always neglected. The latter is consider- able when the system configuration is asym- metrical or the phase load is unbalanced. The conventional approach which considers only the magnitude portion may therefore distort the explanation of the calculated results and lead to incorrect conclusions. The paper explores the effects of the argument portion as well as magni- tude portion on line loss. Examples are given to demonstrate the effects of the phase load unbal- ance and the significance of the argument portion on the line loss.

1 Introduction

A three-phase AC power system is designed to be bal- anced to perfect the utilisation of three-phase com- ponents and loads. However, intrinsic asymmetry in the configuration of power-system components such as untransposed transmission lines, open wye and open delta transformers make the system unbalanced [l-31. The predominant cause of unbalance is unbalanced single-phase loads. Important single-phase loads include AC railway supplies and single-phase furnaces. These large single-phase loads, especially the high-speed railway traction motors which draw considerable unbalanced currents from the power-supply system, lead to the system voltage and current being unbalanced [4, 51.

Currently, some high-speed railways are under con- struction and are in the planning stage. We expect that more high-speed railroads will appear in the world in the near future. The additional losses of components in power-supply systems due to the large single-phase trac- tion loads should be accurately evaluated to confirm the feasibility of these high-speed railway projects. Unbal-

Q IEE, 1995 Paper 1708C (P9), first received 20th April 1994 and in revised form 27th October 1994 The author is with the Department of Electrical Engineering and Tech- nology, National Taiwan Institute of Technology, Taipei, Taiwan 106, Republic of China

I E E Proc.-Gener. Transm. Distrih., Vol . 142, No. 2, March 1995

anced loads also result in the system components becom- ing derated [6, 71.

In this paper, the complex unbalance factor is adopted for rigorous evaluation of line loss due to load unbalance. The definition of complex unbalance factor is therefore introduced first. An analytic method is then proposed to evaluate the power loss of transmission or distribution lines operating in an unbalanced condition. Examples are then given to demonstrate the effect of load unbalance on the line loss.

2 Complex unbalance factors

The degree of unbalance is usually defined, using the method of symmetrical components, by the ratio of the negative-sequence (or zero-sequence) component to the positive-sequence component [8, 91. The amount of voltage unbalance is expressed in symmetrical com- ponents as:

voltage unbalance factor

(1) negative-sequence voltage positive-sequence voltage

-

This is the most precise and meaningful definition of voltage unbalance. However, it is difficult to measure and calculate these sequence components without special instrumentation. Some utilities instead use an approx- imate expression. The voltage unbalance is calculated approximately as follows [8,9] :

voltage unbalance

(2) maximum deviation from average voltage

average voltage

where voltages are measured from phase to neutral, and average voltage is defined to be the average of the magni- tudes of the three-phase to neutral voltages. This is the simplest method of expressing the voltage unbalance.

Although the voltage unbalance with the approximate expression can be easily obtained, a considerable error will be incurred using this approximate definition in some studies. For the purposes of this paper, the precise definition of current unbalance is adopted. Therefore, the power losses of transmission lines and distribution feeders can be rigorously evaluated and the effects of load unbalance on the transmission line loss become obvious.

The negative-sequence (or zero-sequence) voltages in a network mainly result from the negative-sequence (or zero-sequence) currents of unbalanced loads flowing in the network. Evaluate the power loss in transmission lines and distribution feeders, applying the sequence cur- rents is more straightforward than using the sequence voltages. The current unbalance factors therefore are

-

173

required. The current unbalance factors are defined in a similar way as the voltage unbalance factors as follows:

2.1 Negative sequence

(3)

where M , = m2LB2 is the complex unbalance factor for

negative-sequence current I , = j I, I LS, is the negative-sequence current I , = I I, I ~ 6 , is the positive-sequence current

and

LB2 = L6, - 6, ( 5 ) The magnitude of the complex unbalance factor, m 2 , is usually used to indicate the degree of current unbalance. Relatively little attention has been paid to the effects of the argument portion, ~ 8 , . This paper represents a detailed study into the effects of the argument portion as well as the magnitude portion on the power loss of trans- mission lines and distribution feeders. 2.2 Zero sequence Likewise, the zero-sequence version of the complex current unbalance factor can be written as

where

M O = mo LBO is the complex unbalance factor for zero-sequence current

I o = I I , I ~6~ is the zero-sequence current and

1101

O - 1 1 1 1 m --

3

Fig. 1 shows the circuit model of a transmission line. The resistance of the transmission line is relatively small com-

Power loss of a transmission line

L zaa

v; Fig. 1 Circuit model ofa transmission line

pared with the inductive reactance. In calculating power flows, bus voltages, and branch currents, the resistances have a minor effect compared with the inductive reac- tances, especially in planning-type calculations. Hence, in some calculations, they are neglected. However, in other calculations they are of vital importance, examples of which include the calculation of line thermal limits and the problems of economic operation. Both are related to the real power loss of the transmission line. The resist- ance is more important when performing these operating- type calculations under three-phase load unbalance or network configuration asymmetry conditions. The real

174

power loss can be calculated using basic circuit theory and will be discussed in detail in this paper.

The most accurate n-equivalent lumped circuit is usually used when dealing with long transmission lines of length greater than approximately 150 miles. For medium-length lines, in the range of approximately 50 to 150 miles, the simpler nominal n-equivalent circuit may be used instead. For a short line of 50 miles or under, the shunt elements may be neglected, making the circuit even simpler. However, attention is paid only to the power losses of equivalent circuit series elements of the lines; the shunt elements are thereby neglected for all transmission lines in this paper.

The relationship between voltages and currents at the terminals of the transmission line can be derived using the three-phase model as follows:

AVabr = Zabc I o b c (9)

= t b r - I / b b c (10)

where

zab zm

' o b c = [E:; z:: Z.1 = series impedance matrix (11)

Inbc = [I. I, I J T = load currents (12)

t b c = [V, V, V,]' = sending-end voltages (13) Vbbc = [ V i V i V;]' = receiving-end voltages (14)

Complex power loss of a transmission line can therefore be represented as

and

S = AVTbc I,*,, (15)

= 'Tbbc ' T b c 'Zbbr (16)

Substitution of eqn. 9 into eqn. 15 yields

Transferring this to the symmetrical-components expres- sion, using the transformation matrix T, yields

s = G 1 2 = G 1 * ~ : 1 2

or

z o o 20, 2 0 2 1: s = [ I o 1, 121[;;: 2:: ;;][;;I

2 2

= c X Z k , L V (17) k = O 1=0

where r l 1i

Because of the symmetry of the series impedance matrix, eqn. 11 can be rewritten as

' 0 , ' a b 'CO

Znbc = 1;:; z:: z:j (21)

I E E Proc.-Gener. Transm. Distrib., Vol. 142, N o . 2, March 1995

Therefore, Z o 1 2 can be found from eqn. 20:

z o o zo, ZO,

z o , ZZl 2 2 2

z,,, = [ Z O 2 Z l , Z12] (22)

in which

Z O O = i(zcw + z b b + Z c c ) + f ( z & + z b , + z&7) z l l = z 2 2 = f ( z a a + Z b b + z c c ) - i ( z o b + zbc + Zm) z O 1 = & z o o + a 2 Z b b + a z c c ) - ; (azo , + z b , + a 2 z c n )

z O 2 & z a a + a z b b + a 2 z < c ) - ;(a2Z,b + z b , + azco) z l 2 = i(Z.0 + a 2 Z b b + aZ,-,) + #(azab + z b , + a2zco)

z 2 1 = + a Z b b + aZZ, , ) + $ ( a 2 z & + z b , + a&,) (23)

The perfectly transposed transmission line is z,, = z,, = z,, = z, z,, = Zb, = z,, = z,

z,, = z,, = z , , = Z Z 1 = 0

(24) Therefore

and Zoo = Z , + 22, I Z , = zero-sequence impedance Z , , = Z , - Z , 3 Z , = positive-sequence impedance

Z , , = Z , - Z , Z , = negative-sequence impedance 4 Effects of load unbalance

In studying the effects of load unbalance, the positive- sequence load current is usually assumed to be 1 per unit. That is, the rated capacity (current) of the line. Also, the power factor is assumed to be 1.0. That means the trans- mission line supplies the only real power to the load. Hence

(25) I , = 1.0 P.U. = I,,,,ed A From eqns. 3 and 6

I , = I I M o I , = I l M 2 The effect of load power factor on the resulting unbal- ances can be neglected. However, the improvement of power factor can reduce power costs, release electrical capacity of the power-transmission system, raise the voltage level, and reduce the line losses. It is generally economical to improve the power factor to near 1.0 p.u. to obtain these advantages. 4.1 Untransposed transmission line Substitution of eqn. 26 into eqn. 17 yields

1 = I: z o o m : + Z I l ( 1 + m:) + ZOl(M0 + M2 MX) + Z o 2 ( M ; + MOM:) + Z , , M : + Z 2 , M 2

(27) [

Taking the real part of eqn. 27 gives the real power loss P = 1:

1 Ro,~:+Rl,(1 + m 3 + WO, + RO2)Cm0 cos (Bo) + mom2 cos (8, - Boll + WO, - Xol)[mo sin (Bo) + mom2 sin (B2 - Boll +(RI, +R21)mzcos(P2)+(X12-X~l)m2sin(B2)

x [ (28) Eqn. 28 is a general form of the real power loss of an untransposed transmission line. This equation indicates

I E E Proc.-Gener. Transm. Distrib., Vol. 142, No. 2, March 1995

that the real power loss is a function of both negative- and zero-sequence unbalance factors. In addition, the equation shows that the power loss is also affected by the arguments of both negative- and zero-sequence complex unbalance factors.

If any end of the transmission line is ungrounded, the zero-sequence current is blocked. Hence, mo = 0. The power loss equation, under this operating condition, can be greatly simplified as

Eqn. 29 indicates the minimum loss which occurs at

8, =tan-'(-) for R I , + R,, # O (30) RI, + RZl

8, = 2nn for RI, + R,, = 0 (31) in which n = 0, f 1. f 2, . . . . 4.2 Completely transposed transmission line For a completely transposed transmission line, Z O l 2 is a diagonal matrix; that is, all the off-diagonal terms in the matrix are zero. Hence, the power loss of a completely transposed transmission line will be

P = I:[Rom: + Rl(l + m i ) ] (32) Eqn. 32 shows that for a balanced load the power loss will be I:Rl, and for an unbalanced load, an additional loss, I:[Romi + Rim:], will occur.

If no zero-sequence current exists,

P = I:[R,(l + mi)] (33) Eqns. 32 and 33 indicate that the power loss of a com- pletely transposed transmission line depends only on the magnitude of the complex unbalance factor. It is not affected by the argument portion.

5 Case studies

A physical three-phase untransposed transmission line in the Taipower system was used to demonstrate the effect of load unbalance. This 60 Hz 345 kV single-circuit, bundle-conductor line has two subconductors per bundle at 18 in bundle spacing and is 13 km long. The series impedance matrix of this line, as shown below, was obtained by a field test carried out by the Taiwan Power Company.

0.8823 + j6.6458

0.6810 + j1.9113

0.6665 + j2.6666 Zobc = 0.6665 + j2.6666 0.9133 + j6.5022

0.6892 + j2.4783

0.6892 + j2.4783 R 0.6810 +j1.9113

0.9573 + j6.2770 Therefore, the corresponding sequence impedance matrix can be found by using eqn. 20 as follows:

1

1

r [

2.2754 + j11.1791

0.2602 + j0.0392

0.2602 + j0.0392 Z , , , = -0.3059 + j0.0054 0.2387 + j4.1229

0.3637 + j0.2073

-0.3784 + j0.2160 R -0.3059 + j0.0054

0.2387 + j4.1229 The positive-sequence current of the line is assumed to be operating at 913 A (rated ampacity) for all cases.

175

5.1 Untransposed transmission line

5.1.1 m, = 0: Assuming m2 = 0, eqn. 28 can be simplified to

225

1 8 0 -

$135-

2 - E 9 0 -

The effect of current (load) unbalance on the line loss can therefore be easily calculated by eqn. 34. The results are shown in Fig. 2. Fig. 2 shows the effects of the magnitude as well as the argument portion on the line loss.

-

1951 I J -200 -1 00 0 t 00 200

BO

Z 324

Ln

- L

0,

2 1 6 4 -

8 4

Fig. 2 m, =

- x - 2% -.- 4% -A- 6% -0- 8% -A- 10%

Line losses ofan untransposed transmission line (when m2 = 0)

~

In Fig. 2 the interesting portion is magnified for clearer viewing. Fig. 3 shows the original size. The effect of load unbalance on the line loss can also be recognised from this Figure.

404-\ ~ . - -- i

$244-- . X - X - Y - x - x d - ~ X ~ ~ b X ~ ~ - ~ - x - x - x - x - x - x

-

01 I I I I -200 -1 00 0 100 200

BO Fig. 3 m, =

--x- 2% -.- 4% -A- 6% -0- 8 % -A- 10%

Line losses ofan untransposed transmission line (when m2 = 0)

~

5.1.2 m, = 0: Assuming mo = 0, the effect of current (load) unbalance on the line loss can be easily calculated by eqn. 29. The results are shown in Figs. 4 and 5.

Fig. 4 shows that for a slightly unbalanced load the total loss may less than that of the balanced load, but for a greatly unbalanced load the loss will be much more than that of the balanced case.

Figs. 3 and 5 show that the additional loss due to load (current) unbalance is small compared with the total loss

176

if the degree of unbalance is small (less than 10%). However, if the degree of unbalance is great, significant additional loss may incur. Fig. 6 shows the consequences of large load unbalance.

203r

198 l I 1 I J -200 -100 0 100 200

Line losses ofan untransposed transmission line (when mo = 0) 4,

Fig. 4 mi = -.- 0 -0- 270 -*- 4% -0- 6% -A- 8% -A- 10%

6 1 2 4

Q

44 t 1 I I 0 100 2 00

8 2 . - \ b o

Fig. 5 m, = -m- o -0- 2% -*- 4% -0- 6% -A- 8"A -A- 10%

Line losses ofan untransposed transmission line (when m, = 0)

41 I I I I -200 -100 0 100 200

Az, Fig. 6 m, =

-x- 20% -.- 40% -A- 60% -0- 80% -A- 100%

Line losses ofan untransposed transmission line (when m, = 0)

~

I E E Proc.-Cener. Transm. Distrib., Vol. 142, No . 2, March 1995

5.1.3 /3, = 0 and /3, = 0: Assuming j3, = 0 and 8, = 0, the effect of current (load) unbalance on the line loss can therefore be easily calculated by the following equation:

The results are shown in Fig. 7.

m2

Fig. 7 and & = 0) In, =

- x - 20% -.- 40% -A- 60% -0- 80% -A- 100%

Line losses of an untransposed transmission line (when 8, = 0

~

5.1.4 MO # 0 and M, # 0 (actual load unbalances): For unbalances reported for lines feeding actual loads, the total line loss where there are negative- and zero- sequence currents at the same time, that is MO # 0 and M , # 0, is a function of m, , m, , j3,, and 8, as depicted in eqn. 28. The effect of load unbalance on the line loss can be easily evaluated by using this equation.

5.2 Completely transposed transmission line For a completely transposed transmission line, all the off- diagonal terms will be zeros. The sequence impedance matrix of the above Section will be modified by setting all the off-diagonal terms to be zero to simulate the trans- posed effect. The sequence impedance matrix for the sample transmission line which is completely transposed is therefore

2.2754 + j11.1791 0 0.2387 +j4.1229

0

1. [ : 0

2012 =

0 0.2387 + j4.1229

5.2.1 m2 = 0: Assuming m, = 0, eqn. 32 can be simplified as

(36) P = I:[R, mi + R , ] The effect of current (load) unbalance on the line loss is shown in Fig. 8.

5.2.2 m, = 0: Assuming rn, = 0, the effect of current (load) unbalance on the line loss can be easily calculated by eqn. 33. The results are shown in Fig. 9.

I E E Proc.-Gener. Transm. Distnb., Vol. 142, No. 2, March 1995

5.2.3 MO # 0 and M, # 0 (actual load unbalances): Total line loss for a completely transposed transmission line where there are negative- and zero-sequence currents at the same time is a function of rn, and m, as depicted in eqn. 32. The results are shown in Fig. 10.

Figs. 8, 9 and 10 show that the power loss of a com- pletely transposed transmission line is not affected by the argument portion of the complex unbalance factor. It is affected by the magnitude portion only.

x-x-x-xi(-x-x-x-x-x-x-x-x-x-x-x-x-x-x I I I 0 100 200

Line losses of a transposed transmission line (when m2 = 0) BO

Fig. 8 no =

-x- 2% -.- 4% -A- 6% -0- 8% -A- 10%

~

lg7 t 1961 I I I I

-200 -100 0 100 2 00 b2,

Fig. 9 "* = -.- 0% -0- 2% -*- 4% -0- 6% -A- 8% -A- IO"%

Line losses o f a transposed transmission line (when m, = 0)

Fig. 10 Line losses o f a transposed line (when M O # 0 and M , # 0)

177

The investigated transmission line is a real line and currently operated in the Taipower system. There are some measured line losses reported. Unfortunately, the magnitudes and phase angles of the corresponding three individual phase currents were not recorded simulta- neously. Hence, the measured losses cannot be directly compared with the calculated losses presented here. However, the measured values are, as expected, all within the reasonable ranges that are evaluated by using the theory presented here. The measured values would not differ appreciably from the calculated values if applied individual phase currents are the same.

6 Conclusions

Simple criteria to estimate the line loss due to load unbalance have been presented in this paper. The complex unbalance factor is used for evaluating the loss in detail. This paper concludes that for an untransposed transmission line the line loss is affected by the argument as well as the magnitude portion of the complex unbal- ance factor. However, for a completely transposed line, the line loss is only affected by the magnitude portion. Furthermore, for a slightly unbalanced load, the total loss may be less than that of the balanced load, but for a

greatly unbalanced load, the loss will be much more than that of the balanced case.

7 References

1 HESSE, M.H., and SABATH, J.: ‘EHV double-circuit untransposed transmission line - analvsis and tests’. IEEE Summer Power Meeting and EHV Conference, Los Angeles, 1970, Paper 70 T P 644- PWR, pp. 984-992

2 CLARKE, G.D., JONES, K.M., and HABIBOLLAHI, H.: ’Phase unbalance and transmission line transposition on interconnected HV transmission systems’, in ’Sources and effects of power system dis- turbance’. IEE conference publication 210, 1982, pp. 93-99

3 CHEN, T.H., and CHANG, J.D.: ‘Open wye-open delta and open delta-open delta transformer models for rigorous distribution system analysis’, IEE Proc. C , Gena . Transm. Distrib., 1992, 139, (3), pp. 221-234

4 WASOWSKI, A.: ‘Voltage asymmetry in a power system caused by ARC furnaces’, in ‘Sources and effects of power system disturbance’. IEE conference publication 210, 1982, pp. 12-17

5 CHEN, T.H.: ‘Criteria to estimate the voltage unbalances due to high-speed railway demands’. IEEEjPES 1994 Winter Meeting, 94 WM 234-5 PWRS

6 BERNDT, M.M., and SCHMITZ, N.L.: ‘Derating of polyphase induction operated with unbalanced line voltage’, AIEE Trans., 1963, 81, pp. 680-686

7 WILLIAMS, J.E.: ‘Operation of 3-phase induction motors on unbal- anced voltages’, AIEE Trans., 1954,73, pp. 125-133

8 IEC Standard 1000-2-1, 1990, p. 35 9 ‘IEEE recommended practice for electric power distribution for

industrial plants’. IEEE Std. 141, 1976, pp. 58-59

178 I E E Proc.-Gener. Transm. Distrib., Vol. 142, No. 2, Morch 1995