Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
Analysis of a three-limb core power transformer under earth fault
M. A. Tsili
S. A. Papathanassiou
Laboratory of Electric Machines and Power Electronics
Electric Power Division, School of Electrical & Computer Engineering
National Technical University of Athens (NTUA)
9, Iroon Polytechneiou Street, 15780 Athens, Greece
tel: (+30)-210-7723658, fax: (+30)-210-7723593, e-mail: st@power.ece.ntua.gr
ZL
Abstract —In this paper, the zero sequence equivalent circuit
is derived for three-limb core (“core-type”) YNyn0 3-phase
power transformers, to take proper account of the low zerosequence magnetizing reactance and neutral earthing
arrangements of the transformer. Subsequently the sequence
component theory is applied to calculate the expected
sequence and phase currents and voltages for a variety of
unbalanced conditions, including single and double phase-toearth faults and open conductor situations. The results can
provide the basis for the detection of zero-sequence flux
conditions in the transformer, that may appear under certain
unbalanced faults which remain undetected by the standard
transformer protections.
I.
NOMENCLATURE
Φk, k=a,b,c
ep0
: Limb magnetic flux, phase k,
: primary winding zero sequence internal
EMF,
es0
: secondary winding zero sequence internal
EMF,
Ip0
: primary winding zero sequence current,
Is0
: secondary winding zero seq. current,
Φlpk, k=a,b,c : leakage flux of primary winding phase k,
Φlsk, k=a,b,c : leakage flux of sec. winding phase k,
ZGP
: primary neutral earthing impedance,
ZGS
: secondary neutral earthing impedance,
ZG
: substation earthing system impedance
(resistance) to infinite earth,
R lpk , k=a,b,c : reluctance of leakage flux path for phase
k of primary winding,
R lsk , k=a,b,c : reluctance of leakage flux path for phase
k of secondary winding,
: reluctance of phase k limb of the
R k , k=a,b,c: magnetic core,
: reluctance of zero-sequence flux (3Φ0)
R0
path external to the core,
Zp
: primary series (short-circuit) impedance,
Zs
: secondary
series
(short-circuit)
impedance.,
: zero-sequence magnetizing reactance,
Z m0
N
Up0
Us0
Rp
Rs
ZS
ZTL
ZT
:
:
:
:
:
:
primary/secondary winding turns ratio,
primary zero sequence terminal voltage
secondary zero sequence terminal voltage
primary winding resistance
secondary winding resistance
positive/negative sequence impedance of
the system upstream the fault position,
: positive/negative sequence impedance of
the transmission line (fault to substation),
: series (short-circuit) impedance of the
transformer (positive sequence),
ZS0
ZTL0
ZPO
ZSO
ZMO
ZL0
Z
Z0
: Positive/negative sequence impedance of
the load (MV side),
: zero sequence impedance of the system
upstream the fault position,
: zero sequence impedance of the
transmission line (fault to substation),
: primary series impedance of the zero
sequence transformer equivalent circuit,
: secondary series impedance of the zero
sequence transformer equivalent circuit,
: shunt impedance of the zero sequence
transformer equivalent circuit,
: zero sequence load impedance (MV
side),
: ZTL+ZT+ZL
: ZTL0+ZT0+ZL0
II.
INTRODUCTION
Large high to medium voltage power transformers are
vital and expensive components in electric power systems
and accordingly high demands are imposed on the study of
their behaviour and the implementation of suitable
protections. Correct representation of the transformer
under unbalanced operating conditions is crucial for the
accurate calculation of fault currents, as well as for the
selection and setting of appropriate transformer and
network protection schemes.
Although sequence transformer equivalents are widely
available in the literature, these models do not always pay
due attention to the type of the transformer magnetic core,
which affects critically its apparent impedance in zerosequence conditions, [1,2]. It is a known fact that in coretype 3-phase power transformers, a high reluctance return
path exists for zero-sequence flux, via the transformer tank
surrounding the core, which results in low values of the
zero-sequence magnetizing impedance, typically of the
order of 50-150%.
In the present paper, the zero sequence equivalent
circuit is derived for a three-limb core YNyn0 3-phase
power transformer, to take proper account of the low zerosequence magnetizing reactance, as well as of the neutral
earthing arrangements. Subsequently the sequence
component theory is applied to calculate the expected
sequence and phase currents and voltages for a variety of
unbalanced conditions, including single and double phaseto-earth faults and open conductor situations. Special
emphasis is placed on the developed zero-sequence flux in
the transformer. The results can provide the basis for the
detection of zero-sequence flux conditions, that may
appear under certain unbalanced faults which remain
undetected by the standard transformer protections. The
motivation for this work has been provided by a case of
Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
severe transformer tank heating, following a permanent
earth fault on the transmission line near the substation,
which went undetected by the existing protections.
III.
DESCRIPTION OF THE STUDY-CASE SYSTEM
A simplified single-line diagram of the studied system
is shown in Fig. 1. It comprises a 150/21 kV YNyn0
transformer fed by an overhead 150 kV transmission line.
Near the substation, one phase of the 150 kV line has been
interrupted and the transformer side conductor has fallen
on the ground, creating thus a combined phase-to-earth and
open conductor fault. The event remained undetected by
the transmission line and transformer protections, resulting
in severe tank heating of the transformer, which was
eventually tripped by the tank temperature protection.
The system and transformer winding connections and
earthing arrangements are shown in Fig. 1. The
transformer under study is a 150/21 kV, 40/50 MVA
(ONAN/ONAF), 3-limb core unit, shown in the pictures of
Fig. 2. Both windings are Y-connected with neutrals
grounded to the substation grounding system, directly at
the HV side and via a 12 Ω/1000 A neutral resistor at the
MV side. The distribution transformers (20/0.4 kV)
connected to the 20 kV distribution network downstream
the power transformer are all delta-connected at the 20 kV
side (vector group Dyn11).
Figure 2. The 150/20 kV, 40/50 MVA, “core-type” transformer.
The main transformer protection relays are indicated in
Fig. 1. Basic transformer protection is the differential
protection (87T), trigerred by the difference of primary and
secondary winding phase currents. A time over-current
400/150 kV
autotransformer
protection relay (51) serves as back up protection for the
HV side. The MV side includes the secondary winding
differential protection (Restricted Earth Fault-87REF), as
well as low-set and high-set ground over-current relays
(resp. EFL and EFH). Time-delayed over-current
protection (51, 51N) is also used for the secondary breaker.
IV.
ZERO SEQUENCE EQUIVALENT CIRCUIT
In the case of 5-limb core (“shell-type”) YY-connected
transformers, zero sequence excitation in one winding
results in negligible current, if the other winding is open
circuited with respect to this sequence, due to the high
magnetizing reactance of the iron core. However, 3-limb
core (“core-type”) units differ fundamentally in this
respect. The resulting zero-sequence flux path is now
external to the core, via the insulating medium (oil), the
tank and metallic connections other than the core, as
shown schematically in Fig. 3. Due to the high reluctance
of this path, the resulting zero-sequence magnetizing
reactance of the transformer is low and therefore a
significant zero-sequence current may result in the excited
winding. To properly account for such situations, a proper
zero-sequence equivalent circuit is required, which should
also take into account the transformer and substation
grounding arrangements.
To derive such a circuit, the transformer representation
of Fig. 3 is used, along with the respective magnetic
equivalent circuit of Fig. 4. Zero sequence voltages are
applied at the terminals of the primary winding, resulting
in a zero sequence flux Φ0 per core limb. From the circuit
of Fig.4 the following relations hold:
N p i p0 − N s i s0 = φ0R a + 3φ0R 0
(1)
N p i p0 = φlpaR a
(2)
N s is0 = −φlsaR a
(3)
Considering that the reluctance of the flux path external
to the core is much higher than the reluctance of the core
iron ( R 0 >> R a ), eq. (1) can be reduced to:
N p i p0 − N s is0 = 3φ0R 0
(4)
150/20 kV substation
DP
LEGEND
Transmission
line fault
150 kV
busbars
O/C
O/C
EFH
Voltage transformer
Current transformer
Circuit breaker (CB)
Protection relay
DP: Distance protection
O/C: Over-current
DIF: Differential
REF: Restricted earth fault
EFH: Earth fault - High
EFL: Earth fault - Low
Measurement
C/B trip command
EFL
REF
DIF
Figure 1. Single line diagram of the system, including the main transformer protections.
Alarm
20 kV
busbars
Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
3φ0
Core
Z GP
3(i p0-i s0 )
3i p0
e+p0
3i s0
e+s0
φ0
φ0
i p0
-
U
Tank
φ0
so
′ = -[Z ′ + 3N 2 Z − 3N(N − 1)Z ]I ′
s
Gs
G s0
′
φlpa
φlpb
φlpc
φlsa
φlsb
φlsc
+ (Z m0 + 3NZ G )(I p0 − I s0 )
i s0
-
where
Z s′ =
Z GS
ZG
⎛
N 2 ⎞⎟
⎜
s
+
ω
R
j
⎟ is the secondary
s
R
2 ⎜
⎜
N
lsa ⎟
⎠
⎝
1
winding series impedance referred to the primary turns.
I p0
Figure 3. Three-phase core-type YNyn transformer under zero
sequence conditions.
Ra
(9)
φb
φa
φc
Rb
U p0
3φ0
Z P0
Rlpa
Npipo
Rlpb
Npipo
Nsiso
Rlsa
Nsiso
Rlsb
Nsiso
Figure 5. Zero-sequence T equivalent circuit of a “core-type” YY
connected transformer, referred to the primary winding.
Rlpc
Ro
From eqs. (7) and (9), the zero-sequence equivalent
circuit of Fig. 5 is readily derived. Its parameters are:
(10)
Z P0 = Z p + 3ZGP − 3( N − 1)Z G
Rlsc
2
Z S0 = Z s + 3N Z GS
Figure 4. Magnetic equivalent circuit for transformer of Fig. 3.
With reference to Fig. 3, the voltage equation for phase
a of the primary winding is the following, where capital
letters are used for phasor notation in the following:
U p0 = I p0 R p + E p0 + 3Z GP I p0 +3Z G (I p0 − I s0 ) (5)
Using eqs. (1) and (4), the zero sequence voltage, Ep0,
induced in the primary can be expressed as:
d(φ0 + φlpa )
e po = N p
⇒
dt
⎡ N p I p0 − N s I s0 N p I po ⎤
E po = jωN p ⎢
+
(6)
⎥
3R 0
R lpa ⎥⎦
⎢⎣
Combining eqs. (5) and (6), referring current Is0 to the
primary winding turns and rearranging terms, it is
eventually obtained:
U = [ Z p + 3Ζ GP − 3(N − 1)Ζ G ]I p0
po
′
(7)
+ (Ζ m0 + 3NΖ G )(I p0 − I s0 )
where Zp= R + jω
p
U s0
Z M0
Rc
Npipo
N
I s0
Z S0
2
p
R lpa
is the primary resistance plus
N
leakage impedance and Zm0= jω
2
p
3R 0
the zero-sequence
magnetizing impedance.
For the secondary winding, using the voltage equation:
U s0 = I s0 Z s + E s0 − 3Z GS I s0 −3Z G (I p0 − I s0 )
(8)
and working in the same way as for eq. (7), it is deduced:
Z M0
− 3N(N − 1)ZG
= Zm0 + 3NZG
(11)
(12)
where the neutral and substation grounding impedances
have been properly incorporated.
Eqs. (7) and (9), along with the equivalent circuit of
Fig. 5, can be per-unitized with the use of the primary and
secondary winding base quantities. In case of transformers
with on-load tap changers or fixed off-load tap positions,
the OLTC position or the off-nominal turns ratio (ONR)
must be taken into account ([3,4]).
V.
ANALYSIS OF UNBALANCED FAULTS
To quantitatively analyse the fault conditions of the
study-case system of Fig. 1, the sequence component
theory can be applied, utilizing the established sequence
equivalent circuits of the various system elements ([1,2,5]).
In Fig. 6 the connection of the sequence equivalents is
shown for the combined earth fault and open conductor
situation, described in Section III. The various impedances
are defined in Section I.
Solving the sequence network equations permits the
calculation of the voltages and currents at the various
points of the network. For instance, the following
expression is derived for the zero sequence voltage Uf0 at
the point of the fault:
U fo =
− 3(Z + Z f )Z Es
0
[(Z + Z s )(3Z f + Z0 ) + (Z + Z f )(6Z0 + 4Zs0 )
(13)
+ 2Zs0 (Z0 + Z f ) + 2Z(Z0 + Z s )]
Hence, the voltage along the shunt impedance, ZM0,
which provides a measure of the zero-sequence flux in the
transformer, is given by:
Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
If
Is1 Zs
Is1 Zs
If
IT1 ZTL
ZT
Es
1:1
ZL
IT2
Uf1
Es
Is2 Zs
ZTL
ZT
Zs0
Is0
ZTL
ZP0
1:1
ZM0
ZL0
Figure 6. Connection of the sequence circuits for the study case
fault (combined earth fault and open conductor
conditions), [2].
f0
Z M0 Z m0
⋅
Z 0 Z M0
(14)
Es
ZP0
ZS0
ZL0
ZM0
Z0
Es
2(Z 0 + Z s0 ) + (Z + Z s )
Interruption of two phases
Z0
Es
U fo =
(Z 0 + Z s0 ) + 2(Z + Z s )
Single phase to ground fault
Z ⋅ ( Z s0 //Z 0 )
Es
U fo =
(Z + Z )(Z s //Z + Z s0 //Z 0 + 3Z f ) + ZZ s
U fo =
ZT
Is2 Zs
IT2
ZL
ZTL
ZT
Uf2
Is0
Zs0
ZTL
IT0
Uf0
ZL
ZP0
ZM0
ZS0
ZL0
(15)
Figure 8.
Connection of the sequence circuits for two open
conductors (phases b and c).
(16)
VI.
(17)
s
Double phase to ground fault
⎤
⎡
⎞
Z //Z
1 ⎛⎜
s0 0
⎟
⎥
⎢
Z s ⎜ Z //Z + 3Z ⎟
⎥ Es
⎢
s0 0
f ⎠
⎝
=⎢
⎥
1
1
1
⎥
⎢ + +
Z
(Z//Z s ) //(Z0 //Z s0 + 3Z f ) ⎥
⎢Z
s
⎦
⎣
IT1 ZTL
Uf1
Interruption of one phase
fo
ZTL
Connection of the sequence circuits for one open
conductor (phase a).
Is1 Zs
To estimate the zero-sequence flux developed in the
transformer under other possible unbalanced conditions,
the same analysis has been conducted for other types of
unsymmetrical faults, occurring at the same position along
the HV in-feed line. Such are the single-phase and doublephase-to-ground faults and the interruption of one or two
phases. The respective connections of the sequence
equivalent circuits are shown in Figs. 7 to 10 and the
relations for Uf0 are the following:
U
Zs0
Uf0
Figure 7.
U m0 = U
ZT
ZL
IT0
Uf0
ZS0
IT0
ZTL
IT2
3Zf
Uf2
If
Is0
ZL
Uf2
1:1
ZL
ZT
Uf1
If
Is2 Zs
IT1 ZTL
(18)
Eq. (14) can then be used to evaluate the zero sequence
voltage and hence flux along the transformer shunt
reactance.
RESULTS AND DISCUSSION
Eqs. (13) and (15)-(18) can be further simplified to
derive practical rules of thumb regarding the expected zero
sequence voltage magnitude. For the typical parameter
values encountered in HV power systems, it holds in
general that Zs<<ZTL<<ZT<<ZL and Zs0<<ZTL0<<ZT0<<ZL0.
Hence, eq. (13) can be reduced to:
Es
(19)
U fo = −
3
For typical HV/MV transformer parameters and fault
position relatively close to the substation (small ZTL0), eq.
(14) yields Umo ≈ 0.9Ufo. Therefore, in the case of the
combined earth fault and open conductor situation, the zero
sequence voltage along the transformer magnetizing
impedance is approximately equal to 0.3 p.u., which is also
a rough indication of the zero sequence flux per limb.
Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
If
Is1 Zs
IT1 ZTL
ZT
Uf1
Es
Typically Zs0 ≈ k Zs, with k=3÷6. For a mean value of
k=4.5, Umo is eventually found ≈0.6 p.u.
In the case of the double phase to ground fault, eq. (18)
is reduced to:
0.5Z s0
Es
(23)
U fo =
Z s0 + 0.5Z s
If
ZL
If
Is2
Zs
IT2
ZTL
ZT
Uf2
Is0
Zs0
3Zf
ZL
If
ZTL
ZP0
ZS0
IT0
Uf0
ZL0
ZM0
which results in Umo approximately equal to 0.4 p.u.
More accurate results regarding the currents and
voltages in the system are easily derived by solving the
sequence networks for each fault. For instance, for the
combined earth fault and open conductor situation of
Section III, the results are shown in Fig. 11, where the
phasors of HV and MV side voltages and currents are
illustrated. Initially the transformer operates with a 30
MVA, 0.9 ind. p.f. load. The calculated magnetizing
branch zero sequence voltage Um0 is equal to 26.95 kV.
Notable, the approximate solution of the previous Section
resulted in Um0 ≈0.3 p.u. i.e. 0.3⋅(150/√3)=26 kV, very
close to the actual value.
Figure 9. Connection of the sequence circuits for a single phase to
ground fault.
If
Is1 Zs
If
IT1 ZTL
ZT
Uf1
Es
Is2 Zs
ZL
IT2
ZTL
ZT
Uf2
Is0
ZL
Zs0
ZTL
ZP0
3Zf
ZS0
IT0
Uf0
ZM0
ZL0
Figure 10. Connection of the sequence circuits for a double phase
to ground fault.
Following a similar line of reasoning, the simplified
equation for Ufo, in case of interruption of one phase, is:
Z0
(20)
U fo =
Es
2Z + Z 0
Taking into account that Z ≈ Z0, Ufo is then approximately
equal to 0.33 p.u. and Um0 approximately equal to 0.3 p.u.
For two open conductors, Ufo is given by:
Z0
(21)
U fo =
Es
Z + 2Z 0
which also results in Um0 ≈ 0.3 p.u.
For the single phase to ground fault, eq. (17) can be
reduced to:
Z s0
Es
(22)
U fo =
2Z s + Z s0
Figure 11. Phasor diagram of the phase voltages (in kV) and
currents (in kA) after the occurrence of the fault.
The results of Fig. 11 correspond to distance of the
fault from the substation equal to 0.5 km. In order to
examine the impact of the fault distance on Um0 and thus
on the zero sequence flux, the calculations have been
repeated for greater lengths of the transmission line
Paper presented at the 16th International Conference on Electrical Machines, ICEM 2004, Cracow, Poland,
September 5-8, 2004.
between the fault position and the substation. Fig. 12
illustrates the calculated variation of Umo with respect to
the fault-to-substation distance, for the five cases of
unbalanced conditions examined in the previous Section.
In Tables I and II, the resulting voltages Ufo and Umo
are presented for the loaded and unloaded transformer,
again for the same five types of unsymmetrical conditions.
TABLE I
TRANSFORMER ZERO SEQUENCE VOLTAGE FOR DIFFERENT TYPES
OF UNSYMMETRICAL CONDITIONS (LOAD 30 MVA, PF=0.9 IND.)
Fault
Uf0 (kV)
Um0 (kV)
Earth fault and one open phase
One open phase
Two open phases
Phase to ground fault
Double phase to ground fault
29.26
29.55
26.67
63.62
40.9
26.95
27.23
24.53
58.61
37.68
Umo (kV)
TABLE II
TRANSFORMER ZERO SEQUENCE VOLTAGE FOR DIFFERENT TYPES
OF UNSYMMETRICAL CONDITIONS (UNLOADED TRANSFORMER)
Fault
Uf0 (kV)
Um0 (kV)
Earth fault and one open phase
One open phase
Two open phases
Phase to ground fault
Double phase to ground fault
29.45
0.24
0.12
63.87
41.11
27.14
0.22
0.11
58.84
37.87
[2]
[3]
[4]
[5]
20
40
60
80
CONCLUSIONS
In this paper, the zero sequence equivalent circuit of a
core-type 3-phase YNyn power transformer is derived,
taking into account the core structure and the winding
earthing arrangements. The equivalent circuit is then
applied for the analysis of various unsymmetrical operating
conditions. The analysis was prompted by an incident
where an unbalanced fault, which remained undetected by
the system and transformer protections, resulted in severe
tank heating of the transformer due to the zero sequence
flux circulating through the tank. The analysis is focused
on the zero sequence voltage and hence flux in the
transformer and can serve as the basis for the elaboration
and implementation of a suitable protection.
[1]
REFERENCES
P. M. Anderson, Analysis of Faulted Power Systems. IEEE
Press, 1995
Electrical Transmission and Distribution Reference Book,
Westinghouse Electric Corporation, 1964.
S. A. Papathanassiou, “Modeling Transformers with OffNominal Ratios for Unbalanced Conditions,” IEEE Power
Eng. Review, pp. 50-52, Feb. 2002.
L. V. Barboza, H. H. Zurn, R. Salgado, “Load Tap Change
Transformers: A Modeling Reminder,” IEEE Power Eng.
Review, pp. 51-52, Feb. 2001.
P. Kundur, Power System Stability and Control. McGraw
Hill, 1994.
100
Distance of substation from the postion of the fault (km)
earth fault and one open phase
one open phase
two open phases
phase to ground fault
double phase to ground fault
Figure 12. Variation of Umo with the fault-to-substation distance
for various types of unsymmetrical conditions.
Comparing the results shown in Tables I and II leads to
the following conclusions:
ii)
VII.
VIII.
65
60
55
50
45
40
35
30
25
20
15
0
i)
provide a low reluctance return path for the flux.
When significant load exists, the closed secondary
winding forces the zero sequence flux to partially
follow paths external to the core. The Umo magnitudes
for the other kinds of faults are similar as for the case
of the loaded transformer.
iii) The values of Table I (loaded transformer) are very
close to the simplified estimations of Um0 provided in
Section V.
For a loaded transformer, the greatest values of Umo
occur in the case of single and double phase to
ground faults. Open conductors result in similar and
relatively lower values.
Under low load conditions, zero sequence voltage and
therefore flux in the transformer is negligible in case
of single- or two-phase supply. This is expected
because the core limbs of the interrupted phases
IX.
APPENDIX: PARAMETER VALUES
The system short-circuit capacity at the point of the fault is
3200 MVA and the X/R ratio is equal to 7. The distance
from the substation to the point of the fault is 0.5 km. The
other parameters of the study case system have the
following values:
0Ω
ZGP =
ZGS =
12 Ω
ZG
=
0.5 Ω
Zm0 =
540j Ω
ZS
=
0.994+6.961j Ω
ZTL =
0.097+0.391j Ω/km
ZT
=
90j Ω
ZL
=
675+326.92j Ω
ZS0 =
5.779+34.678j Ω
ZTL0 =
0.497+2.349j Ω/km