Analysis of 3-Phase Symmetrical and Unsymmetrical Fault on
Transmission Line using Fortescue Theorem
FSAHA MEBRAHTU GEBRU1, AYODEJI OLALEKAN SALAU2,5,
SHAIMAA HADI MOHAMMED3, S. B. GOYAL4
1Department of Electrical and Computer Engineering, Haramaya University, Haramaya, ETHIOPIA
2Department of Electrical/Electronics and Computer Engineering, Afe Babalola University,
Ado-Ekiti, NIGERIA
3Department of Computer Science, Sumer University, IRAQ
4Faculty of Information Technology, City University, Petaling Jaya, 46100, MALAYSIA
5Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, INDIA
Abstract: This paper investigates the major faults affecting the transmission of electrical energy after power has
been generated from the power generating station. In a 3-phase transmission line, faults arise due to numerous
causes such as aircraft, line breaks due to the excessive loading, heavy winds, trees falling across the lines etc.
The faults faced in a 3-phase transmission line are broadly categorized into two main parts, namely:
unsymmetrical faults and symmetrical faults. Furthermore, there is another classification of faults in 3-phase
transmission lines such as: shunt type of faults and series type of faults, but this paper discusses the shunt type
of faults which create short circuit on single line to ground (L-G) faults between two conductors or line to line
(L-L) faults, or double line to ground (LL-G) or (triple) three line to ground (LLL-G) faults. This was achieved
using the Fortescue Theorem on MATLAB software. The results show that the single L-G faults occur more
frequently followed by the L-L faults, LL-G faults, and LLL-G faults. This study is essential to evaluate the
power reliability and stability of power transmission lines.
Keywords: Transmission line, Line to ground, Double line ground, three line to ground, 3-phase fault
Received: July 14, 2021. Revised: July 25, 2022. Accepted: September 21, 2022. Published: October 19, 2022.
1 Introduction
Electric power is transmitted and distributed
through transmission lines. In many cases, the
voltage level of the transmission line is raised by a
transformer before electric power is transmitted
through the transmission line. Basically, electric
power is proportional to the product of current and
voltage, in transmission lines, high voltage is
transmitted in order to reduce the line current i2r
losses. Operation of a 3-phase AC power system
has equal distribution of current and voltage
magnitudes in each phase when operated in normal
condition; however, faults may occur to disrupt this
condition. The types of faults created in a
transmission line may be balanced or unbalanced.
Balanced faults involve all phases while
unbalanced faults involve only 1 or 2 phases.
Unsymmetrical and symmetrical fault analysis is
performed to determine the value of the fault
current in KVA or in MVA, [1]. Faults in
transmission lines are caused by circuit failure
which interferes with the normal flow of current. It
is the undesirable creation of conducting path for
short circuit or open circuit fault which blocks the
flow of current, [2]. When a fault occurs in a
transmission line, the short circuit current is high,
usually six to ten times more than the normal full
load current in the system, [3]. The growth of
power systems with increasing load demand has
brought about the need for speed and accuracy of
power transmission equipment. Transmission line
faults which are not detected early and removed
cause blackout or wide spread damage of power
system equipment, [4].
This paper presents the analysis of a 3-phase
transmission line with resistive, inductive, and
capacitive loads (RLC) during the L-G faults, L-L
faults, LL-G faults, and 3-line to ground (LLL-G)
fault.
2 Proposed Technique
2.1 Proposed Test System
In this section, we present the test system which
consist of a 3-phase voltage source and 3-phase
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2022.17.32
Fsaha Mebrahtu Gebru, Ayodeji Olalekan Salau,
Shaimaa Hadi Mohammed, S. B. Goyal
E-ISSN: 2224-350X
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load as shown in Figure 1. The loads active and
reactive power variation with the positive sequence
component of voltage are given by (1) and (2), [5].
0 1 2
0
( ) (1 )(1 ) (1)
np pp
V
p p T R
V
where P is active power, Po is reference active
power, Tp1 and Tp2 are time constants, V is the
voltage, and V0 is the reference voltage, [6].
0 1 2
0
( ) (1 )(1 ) (2)
nq qq
V
Q Q T T
V
where Q is reactive power, Qo is the reference
reactive power, Tq1 and Tq2 are time constants, V is
the voltage, and V0 is reference voltage. In (1) and
(2), nq = 1 if V > Vmin and nq = 2 if V<Vmin.
Fig. 1: Test System for the Study of Fault Analysis
a. Unsymmetrical and Symmetrical Fault
Analysis using Fortescue’s Theorem
A symmetrical fault is a fault where all the 3-
phases are affected equally, thereby making the
system balanced. This type of fault usually arise
from symmetrical currents or short circuit currents,
[7]. Figure 2 gives an illustration of a current fault
in a 3-phase line in a short circuit condition. Hence
only positive sequences are needed to analysis the
fault, [8].
Fig. 2: Symmetrical fault.
Any three unbalanced set of voltage or current can
be resolved into three balanced sets of voltage or
current. This condition indicates that the system is
symmetrical.
Positive sequence components: three phasor of
equal magnitude displaced by 120 degree from
each other following the positive sequence.
Negative sequence components: three phasors of
equal magnitude displaced by 120 degree from
each other following the negative sequence.
Zero sequence components: three parallel phasors
having the same magnitude and angle.
For 3-phase systems, three unbalanced phases can
be resolved into three balanced systems of three
phasors each.
The transmission 3-phase voltage can be expressed
as Va, Vb, and Vc. According to the Fortescue
Theorem these can be transformed. Positive
sequence voltage are supplied by the power
generator within the system and are always present
and are displaced 120 degree apart from the other
lines, but display a counter clockwise rotation
sequence of A-B-C. The representation of positive
voltage is expressed in terms of Va1, Vb1, and Vc1,
while negative sequence voltage is expressed in
terms of Va2, Vb2, and Vc3, and zero sequence are
given as Va0, Vb0 and Vc0.
Thus: Va = Va1+Va2+Va0
Vb = Vb1+Vb2+Vb0
Vc = Vc1+Vc2+Vc0
Figure 3 shows the representation of positive,
negative, and zero sequence voltages.
(a)
(b)
(c)
Fig. 3: Representation of (a) positive, (b) negative,
and (c) zero sequence.
The ‘a’ operator of 3-phase transmission line is
given as:
a = 1< 1200 = -0.5-j0.866
a = |1| rotates by 1200
a2 = 1< 2400 = -0.5-j0.866
a3 = 1< 3600 = 1< 00 = 1+j0
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Fsaha Mebrahtu Gebru, Ayodeji Olalekan Salau,
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Therefore this summation is 1+a+a2= a3+a2+a.
Figure 4 presents the representation of the ‘a’
operators for the 3-phase transmission line.
Fig. 4: ‘a’ operators of transmission line.
From the above equations,
Vb1 = a2Va1, Vc1 = aVa1,
Vb2 = aVa2, Vc2 = a2Va2,
Vb0 = Va0, Vc0 = Va0.
Finally, the positive sequence voltage is given as:
Va = Va0 + Va1 + Va2
Vb = Va0 + a2Va1 + aVa2
Vc = Va0 + aVa1 + a2Va2
This relationship is given in matrix form as:
Then the matrix expression is given as:
Therefore, Vp can be expressed in matrix forms:
Vp = AVs and Vs = A-1Vp
Va1 = 1/3(Va + aVb + a2Vc)
Va2 = 1/3(Va + a2Vb + aVc)
Finally the power of the 3-phase using symmetrical
component analysis is given as:
Note that AT = A A* =
In Figure 5, we show the flowchart of our proposed
method which we used for the analysis of
symmetric faults. However, the major faults
created in transmission lines are unsymmetrical in
nature. These include single L-G faults, LL-G
faults, and L-L faults as shown in Figures 6, 8, and
9, respectively. In the 3-phase system, single L-G
faults are observed to occur more frequently
followed by the L-L faults, LL-G faults, and 3-
phase faults. At the time of electrical storms, the
above types of faults do occur, which ultimately
affect transmission lines, [9].
Single L-G faults: Single L-G fault is the most
common type of shunt fault. Single line to ground
fault accounts for 70 to 80 % of faults that occur on
transmission lines which cause interruptions in
power supply, [10], [11].
This fault occurs when a conductor is in contact
with the ground or a neutral terminal. A brief
illustration is given in Figure 6. Suppose that phase
‘a’ is connected to ground at the fault point ‘F’,
where, the fault impedance is given as Zf. By
convention, the fault current is taken as positive
when flowing out of the fault point.
From Figure 6, phase ‘a’ is connected to ground at
the fault, phase ‘b’ and ‘c’ are in open circuit mode
and carry no current.
Therefore Ib = Ic = 0 at ‘F’, Va = ZfIa and the
sequence of current at the fault is given as:
where
Furthermore, Va = IaZf = 3Ia1Zf and
therefore Ia1 =
WSEAS TRANSACTIONS on POWER SYSTEMS
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Fsaha Mebrahtu Gebru, Ayodeji Olalekan Salau,
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E-ISSN: 2224-350X
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Fig. 5: Flowchart of symmetric faults.
Fig. 6: Single L-G fault.
The sequence component connection for the single
L-G fault is shown in Figure 7.
Fig. 7: Sequence component of the single L-G
fault.
Generally, the equation of current is:
,
The summation of these currents is given as:
L-L fault is the second most prominent type of
fault that occurs in a transmission line when the
two lines are shorted together. This is shown in
Figure 8 and further explained in [12], [13].
Fig. 8: L-L fault.
Set bus count i = 1
Start
Read
data
Formulate & store Z1
bus
Set E = 1 < 10 & Zf = 0
Specifies the fault
Find the fault current
Calculate Ia, Ib, Ic, and Ik
Print Ia, Ib, Ic and Ik
Find Va, Vb & Vc
Print Vi1, Vi2, Vi0, Va, Vb, and Vc
Advanced bus count by 1
Stop
Find v1 using vi1
E=Z1ikI1k
Is in
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Fsaha Mebrahtu Gebru, Ayodeji Olalekan Salau,
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The overall line current equations are given as:
1
12
21
00
a
a
aa
a
E
IZZ
II
I
LL-G is the type of fault which occurs when two
phases of a power system fall on the power line.
This is depicted in Figure 9, [14], [15].
Fig. 9: LL-G fault.
The overall value of currents in this configuration
is given as follows:
102
102
2
01
02
01
02
.
2
a
a
aa
a
E
IZZ
ZZZ
Z
II
ZZ
Z
Ia I
ZZ
The flowchart used for the analysis of
unsymmetrical faults is shown in Figure 10.
Fig. 10: Flowchart for the analysis of
unsymmetrical faults.
3 Simulation Results and Discussion
The proposed approach analysed a load connected
to a 3-phase generator using a single transmission
line. The transmission voltage and current is
measured at the load terminals. The outputs of
current and voltage wave forms for all the 3-phases
are provided for each case of the study in the
following sections.
a. Single L-G Faults
A L-G fault is one in which a short circuit occurs
between one phase (line A) of the system and the
earth. The simulation result of the voltage during
the L-G fault is given in Figure 11. The results
show that the value of voltage for the asymmetrical
fault line to ground decreases in amplitude, while
the voltage is stable or rise slowly during the time
of fault occurrence.
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Fig. 11: 3-phase voltages during L-G fault with R-
L load.
In Figure 12, the value of current rises above its
amplitude which produce a high current during the
fault events. The amplitude of the fault current is
observed to increase above the normal per unit
current after 0.4 seconds. In single L-G faults, the
current flows through the ground from the phase
that is faulty. The current drawn by the load during
the faulty conditions is depicted in Figure 13.
Fig. 12: 3-phase current during L-G fault with R-L
load.
b. L-L Fault
A L-L (unsymmetrical) fault is simulated with the
fault between phase A and Phase B. The two
conductors are short circuited and the result is
shown in Figure 13. The value of the L-L voltage
magnitude between phase A and phase B reduces
during the fault conditions.
Fig. 13: 3-phase voltages during L-L fault with R-L
load.
During the L-L fault, the current drown by the load
during the fault with R-L load is depicted in Figure
14 and it observed from this Figure 14 the value of
current is exceeds above the normal conditions.
Fig. 14: 3-phase current during L-L fault with R-L
load.
C. 3-Phase Fault (LLL-G)
The 3-phase fault to ground involving earth has
been simulated by connecting all 3-phases to earth
simultaneously. The resulting waveform of the
voltage is shown in Figure 15. The results show
that all the 3-phases are reduced below the normal
value (0.05 or -0.05pu).
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Fig. 15: 3-phase voltages during LLL-G fault with
R-L load.
In Figure 16, it is observed that the 3-phase current
drawn by the load during the LLL-G fault rises
above the normal value (5 or -5 pu) at different
magnitudes and amplitudes.
Fig. 16: 3-phase current during LLL-G fault with
R-L load.
4 Conclusions
This paper presented the analysis of faults in a 3-
phase transmission line with resistive and inductive
loads. A significant increase in current above the
normal conditions was obtained for asymmetrical
faults. Consequently, the current and voltage wave
forms are reduced. The effects of the voltage wave
forms on 3-phase faults was seen to affect both the
symmetrical and unsymmetrical 3-phase faults.
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DOI: 10.37394/232016.2022.17.32
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Shaimaa Hadi Mohammed, S. B. Goyal
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Volume 17, 2022
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DOI: 10.37394/232016.2022.17.32
Fsaha Mebrahtu Gebru, Ayodeji Olalekan Salau,
Shaimaa Hadi Mohammed, S. B. Goyal
E-ISSN: 2224-350X
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