Phase co-ordinate approach to calculate earth-fault current and shock voltage

C.Y. Teo W.X. He T.W.Chan

Indexing terms: Earth ,fault current, Shock voltage, Eurthing systems, Fuult unulysis

Abstract: The paper summarises an alternative approach for the calculation of earth fault currents and shock voltages for low voltage systems up to lOOOV without using the conventional symmetrical components. As the shock voltage at each section of the installation has to be modelled during fault conditions, the system equations are formulated entirely in the phase co-ordinates including the representation of neutral and grounding conductors. Test results on two different earthing systems demonstrate the ability in modelling the untransposed cable parameters and the influence of the unbalanced load currents on the postfault voltages. More accurate results can thus be obtained as compared with other methods of calculation.

I

1 Introduction

The analysis of unbalanced fault currents in a three- phase network by means of symmetrical components has been developed since 1918 [l]. By resolving into the positive, negative and zero sequence networks, the postfault voltages and currents in each phase of the network can be calculated by means of a transforma- tion from the sequence networks. This traditional method [2, 31 led to a greater understanding of network behaviour, giving rise to subsequent development in the associated technology.

For low voltage (LV) systems of 50V to lOOOV, the supply is provided by a three-phase, three-wire system with protective conductors (PC) which are also known as grounding conductors [4]. The supply can also be provided by a three-phase, four-wire system (i.e. three- phase and neutral) with PC. In either case, it is required that all the exposed-conductive parts which refer to any metallic parts of electrical equipment, other than parts which are normally live, should be connected by PC to a main earthing terminal. The designer should also ensure that during an earth fault, the shock voltage [4], or touch voltage [5] which is the 0 IEE, 1997 IEE P ~ ~ ) ~ ~ ~ A ~ ~ ~ no. io971 42n

Paper first received 4th November 1996 and in revised form 21st April I997 The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798

voltage between the exposed-conductive-parts and earth shall be of such magnitude as not to cause danger [5 , 61. By using a single phase representation, the calcu- lation of the earth-fault current, IF, is shown in eqn. 1 and the shock voltage, VSK, in eqn. 2

I F = - (1) Eo

2s -t z p + ZpC + Z G

where Eo is the line-to-earth voltage, 2, is the source impedance, ZP is the per-phase impedance of the phase conductors, ZPc is the per-phase impedance of the PC and Z, is the grounding impedance at the source of

This single phase representation gives reasonably accurate answer for design purposes. However the mutually coupling effect from the other two healthy phases, voltage rise in the neutral conductor and the contribution of fault current by motors, are not mod- elled. By using symmetrical components, a more accu- rate calculation [4] for a single-line-to-earth-fault can be obtained by eqn. 3:

supply.

( 3 ) 3E1 I F = -

21 + 22 + 20 + 3Zc: where El is the line-to-neutral voltage, Z1, Z, and Z, are the positive, negative and zero sequence imped- ances of the network.

For LV supplies in buildings, most single-core cables are clipped directly on a flat surface touching each other or spaced by certain clearance. They are not equilateral triangularly spaced and are not transposed. Thus the mutual impedance between the three phases are not equal. In other words, the circuit parameters are not balanced. However, the calculation by symmet- rical components is valid only when the circuit parame- ters are balanced or all the cables are to be completely transposed. In some circumstances, the earth fault cur- rents in the LV system may be comparable with the magnitude of the load currents which may be unbal- anced. Although these load currents can be modelled, they have to be assumed as balance in the calculation by using symmetrical components.

By using a single phase representation, the shock voltage at each section of the installation can be calcu- lated by eqn. 2 if the fault current and the impedance of PC at each section are known Using symmetrical components, the PC is combined with the phase con- ductors as 2,. The calculated postfault voltages expressed in sequence values are not related to the shock voltage. After transformation, the postfault

441 IEE Prot -Electr. Power Appl., Vol. 144, No. 6, November 1997

phase voltages are obtained but not the postfault volt- age at the PC i.e. shock voltage.

This paper introduces a direct approach by using only the phase voltage and phase current for the analy- sis of unbalanced fault currents in a three-phase four- wire system with PC. This direct approach by repre- senting the power network in its physical network con- figuration avoids the transformation of variables to the symmetrical components [7-91. By modelling the sys- tem equations of the power network in the phase-coor- dinates including the representation of neutral and grounding conductors, different types of faults can be modelled by a simple modification to the system equa- tions according to the type of fault. The postfault phase voltages, neutral voltage and the voltage at PC can be obtained by solving the n system equations for the n unknown values. There is no need to connect the three sequence networks according to the type of fault [8]. The proposed method of calculation has been dem- onstrated with several test cases for two different earth- ing systems. The results obtained have also been compared with those obtained using eqns. 1 and 2 as well as by symmetrical components.

2 TN-S earthing system

In the TN-S system [6], the source of supply is directly earthed (T) and the exposed-conductive parts con- nected to the installation's earthing terminal are earthed at the neutral point (N) of the supply source through a separate (S) protective conductor as shown in Fig. 1. The main feature of this system is that there is a continuous metallic path from the exposed-conduc- live parts to the neutral of the source and therefore it usually results in a higher earth fault current.

DU D1 D2 D3

'

22 kV 400V 60 kW

- -0.1778 0.0889 0.0889 0.1540 -0.1540 0.0

0.0889 -0.1778 0.0889 0.0 0.1540 -0.1540

0.0889 0.0889 -0,1778 -0.1540 0.0 0.1540

0.1540 0.0 -0.1540 -0.2667 0.0 0.0

-0.1540 0.1540 0.0 0.0 -0.2667 0.0 0.0 -0.1540 0.1540 0.0 0.0 -0.2667 -

Fig. 1 Sample TN-S system ii) 50m 3x95"' CuiPVC jii) 20m 3x25mmz CUIPVC (iii) 50m 1x50mm2 CuiPVC (iv) 20m lxlOmmZ CuiPvC

2. *I Utility system The utility system is the main source of the fault cur- rents. The amount of the short-circuit current from the utility system is normally expressed as a three-phase fault level in MVA with a X/R ratio at the service entrance. In the tested system as shown in Fig. 1, the utility 22kV networlt connected to DU is represented by an equivalent generator with Z , = Z2 = 2,. Based on a fault level of 800MVA with a XIR ratio of 10, the value of Z1 is 0.01244+j0.1244p.u. on a 100MVA, 22kV base. The 6.5Q earthing resistor at the 22kV incoming source is modelled as a resistor connected from the neutral of the equivalent generator to earth. In the phase representation, the sequence matrix Z,,, can be converted to the phase matrix ZABc or YABC [7]. After converted to the phase values, the 4x4 admit- tance matrix YABC of the equivalent generator includ-

442

ing the earthing resistor is:

-0.796+37.96

0.0 0 .796pj7 .96 0.0 -0.796+j7.96

0.796-j7.96 -0.796+j7.96 0.0 0.0

0.796-37.96 0.0 0.0

-0.796fj7.96 -0.796fj7.96 -0.796+j7.96 3.13-223.88 i

_.._ ~.

. . . . -...

5.1 mm 7.98 mm 5.1 mm

hg.2 In a three-phase three-wire system, the neutral conductor IS removed, but the space IS kept

Layout of circuit conductors fvom DI to D2

single-core cable and their mutual reactances are calcu- lated [3, 111. In the phase co-ordinate, the 4x4 imped- ance matrix of the 50-m circuit from D1 to D2 expressed in p.u. values on a 100MVA, 400V base is :

I 1

6.94+j17.78 314.64 j13.28 j11.95

j14 .64 6 .94+~17.78 j14.64 j12.53

j13.28 314.64 6.94+j17.78 213.35

311.95 j12.53 313.35 13.19+318.40 1 where the resistive value of the 95"' phase conductor is 6.94p.u. and the resistive value of the 50mm2 PC is 13.19p.u.. Obviously, it can be seen that the circuit parameters are not balanced since not all off-diagonal elements are identical. Similarly, the 4x4 impedance matrix of the 20-m circuit from D2 to D3 is:

10.55+36.91 j5 .41 j4 .86 j4 .33

j 5 . 4 1 10.55+36.91 j5 .41 j4 .56

j4 .86 j5 .41 10.55+36.91 j4 .90

j4 .33 j4 .56 j4 .90 26.38+37.28

IEE Proc.-Electr. Powev Appl., Vol. 144, No 6, November 1997

2.4 Induction motor The fault current contributed from induction motor is generated by inertia driving the motor in the presence of a field flux produced by induction from the stator and thus, the current contribution of an induction motor in the first or second cycle after the fault should be considered. The 60-kW induction motor is modelled as an un-earthed generator with a subtransient reac- tance of 25% on 75KVA with X , = X , = XO. The value of XI is 333.33p.u. on a 100MVA, 400V base. After converted to the phase co-ordinate [7], the 4x4 admit- tance matrix of the motor represented by an unearthed generator is:

-0 003 0.0 0.0 0.003

-0.003 0.0 ' 1 ::: 0.0 -0.003 ::::: 1 0.003 0.003 0.003 -0.009

2.5 System equations With the admittance matrix representation of the utility system, transformer, cables and motor in the phase coordinates, the system equation can be assembled.

DU

The nodal admittance system equations relates the nodal voltages and currents at every node in the net- work for each of the phase. The order of the system admittance matrix for 1 he network in Fig. 1 or in Fig. 3 is 17x17 which can be used to solve for the 17 post- fault voltages. The po!;t-fault voltages including shock voltages and the current distribution for an earth fault from phase A to the frame of the motor is shown in Fig. 3. The comparison of results calculated by the two conventional methods and the proposed methods is given in Table 1.

To illustrate the flexibility of our approach, it is assumed that the earth fault current blew the fuse at phase A of the circuit from D2 to D3. This phenomena can be modelled by inserting an node 18 as shown in Fig. 4 in which the ciilculated postfault voltages and current distribution are also shown.

3 TT earthing system

In the TT system [6], the source of supply is directly earthed (T) and the exposed-conductive parts connect- ing to the installation's earthing terminal are also directly earthed (T) through its own earth electrode.

D1 D2 D3 227~35 196~37' 155~38'

0 @ 0 ...................................... 12.7 K L ~

i I I I @ 24.4~-48 i

232~90' 232~90'

Fig. 3 Postfault voltages and current distribution for the TN-S system

D1 D2 D3 DU 231~ -30' 230~-30 18~-103~

0 G9 0 ...................................... - #- + OLU

I 18L-102°

.....................................

Fig.4 Fuult current distribution ufleu the operation of protective device ut node 9

443 IEE Proc.-Electr. Poiver Appl., Vol. 144, No. 6 , November 1997

Table 1: Comparison of results for TN-S system

Single Symmetrical phase component

Total fault current (A) 2455L-44" 2505L-42"

T/F current A 2455L-44" 2457L-40" at400V(A) B - 48L-101"

C - 481-101"

TIF current A 25.8L-44" 25.6L-39"

- O L O " a t22V(A) B

C - 25.6L-141"

Shock D3 156L-39" - 55L-27" - voltage at D2

D1 Of 0" - (VI

Proposed method

2388L-5 1

2335L-49"

43L-119"

54L-97"

24.4L-48"

0.22L133"

24.1 L132"

155L-38"

56L-25"

O L O "

EU El E2 E3

22kV 400 V 400V 400V

Fig.5 Sample TT system (1) 50m 4x95mmz CdPVC (11) 50m 4x50mmz CdPVC (111) 50m 1x25mm2 CuiPvC

The earthing at the source of supply is independent of the earthing at the installation. This arrangement is

E l E2

208L-0.3 2 0 3 ~ -2"

shown in Fig. 5. The main feature of this system is that there is no continuous metallic path between the exposed-conductive parts and the neutral of the source. The earth-fault current flows via two earth electrodes and the mass of the earth.

In the tested system as shown in Fig. 5, the utility 22kV network connected to EU and the 22kVl400V transformer are represented by one equivalent genera- tor with XI = X , = X , as shown in Fig. 6. Based on a fault level of 25MVA with a XIR ratio of 5 at El , the value of XI is 0.7845+j3.9225p.u. on a lOOMVA, 400V base. The 50-m circuit from E l to E2 consisting of four identical single-core, PVC-insulated copper conductor cables of 95mm2 without PC, and the other 50-m cir- cuit from E2 to E3 has an additional PC of 25mm2. The layout of the circuit conductors is the same as those shown in Fig. 2. Obviously, the order of the impedance matrix for the circuit from E l to E2 is 4x4 and the circuit from E2 to E3 is 5x5. It is assumed that the unbalanced load can be lumped at E3 as LA = 200A, L, = 250A and Lc = 300A and a power factor of 0.8. It is also assumed that the earth electrode resist- ance at the installation is 0.9Q and that at the equiva- lent generator at E l is 0.19.

For an earth fault from phase A to the exposed con- ductive part at E3 as shown in Fig. 6, the earth fault current is actually less than the load current in the other two healthy phases. The shock voltages are also higher than those in the TN-S system in Section 2. The order of the system equations is 14x14 which can be solved to obtain 14 postfault voltages. The fault cur- rent distribution and the postfault voltages including

E3

198.L-2" i

193L-3 198~-2"

- - Fig. 6 Postfault voltages and current distribution for the TT system

444 IEE Pvoc-Electr. Power Appl., Vol. 144, No. 6, November 1997

shock voltages and neutral voltages are shown in Fig. 6. The comparison of results calculated by the two conventional methods and the proposed method is shown in Table 2.

Table 2: Comparison of results for TT system

Single Symmetrical Proposed phase component method

Total fault current (A) 221~-1" 202~-1" 2 15L-3" Current at A 221L-1" 433L-21" 386L-19" El (A) B - 255L-156" 245L-156"

C - 257L 83" 287L83" N - - 80L122"

Shock volt- E3 204L-2" - 198L-2" ages(V) E2 199L-1" - 193L-3" Neutral E3 - - 22L-176"

- - 22L-173" voltage (V) E2

El - - 21L177"

4 Conclusioli

A direct approach for calculation of earth-fault cur- rents and shock voltages for low-voltage systems is proposed. By modelling the system equations in the phase coordinates including the neutral and grounding conductor, more accurate results can be obtained as compared with other methods such as those using sym- metrical components. Test results on two different earthing systems demonstrate the ability in modelling the unbalance in circuit parameters and the influence of any unbalanced load currents.

For HV networks, the distributed capacitive reac- tance over the length of the transmission line and the

representation of earth wire can also be modelled in the phase co-ordinates [7, 81. Although only delta-earthed wye transformer is described, other winding connec- tions including three-winding transformers can also be modelled by using the same approach [7, 8, 101. In this way, the proposed method can also be used for the analysis of general power network. Thus, it may be possible that in the future years when this proposed approach gains sufficient support in the industry, the terms of symmetrical components may no longer be necessary.

5 References

1 FORTESCUE, C.L.: 'Method of symmetrical coordinates applied to the solution of polyphase networks', Trans. AIEE, 1918, 37, pp. 1027-1 140

2 KIMBARK, E.W.: 'Power system stability, vol 1' (Chapman & Hall, 1952)

3 ANDERSON, P.M.: 'Analysis of faulted power systems' (IEEE Press, 1995)

4 IEEE Std 142-1991: 'IEE recommended practice for grounding of industrial and commercial power system'. IEEE, 1991

5 JENKINS, B.D.: 'Touch voltages in electrical installations' (Blackwell Scientific Publications, UK, 1995)

6 'IEE regulations for eleclrical installations'. 16th edn., IEE, UK, 1991

7 LAUGHTON, M.A.: 'A.nalysis of unbalanced polyphase net- works by the method of phase coordinates. Part I: System repre- sentation in phase frame of reference', Proc. IEE, 1968, 115. (8), pp. 1163-1172

8 LAUGHTON, M.A.: 'Analysis of unbalanced polyphase net- works by the method of phase coordinates. Part 11: Fault analy- sis', Proc. IEE, 1969, 116, ( 5 ) , pp. 857-865

9 LAUGHTON, M.A. and SAY, M.G.: 'Electrical engineer's refer- ence book' (Butterworths, 1993, 15th edn.)

10 CHEN, M.S.: 'Power system modelling', Pvoc. IEEE (USA) , 1974, 62, (7), pp. 901-915

11 GRAINGER, J.J., and STEVENSON, W.D.: 'Power system analysis' (McGraw-Hill, 1994)

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