ELEC 301/401 – Power Generation,
Transmission and Distribution
Example – Fault Analysis
Dr Roberto Ferrero
Email: Roberto.Ferrero@liverpool.ac.uk
Telephone: 0151 7946613
Office: Room 506, EEE A block
Roberto Ferrero – ELEC 301/401 – 2018/2019
1
Unbalanced fault analysis
Problem definition
A three-phase line is connected to an infinite busbar through a star-starconnected transformer with grounded neutrals on both sides (primary side:
busbar – secondary side: line).
The equivalent transformer and line impedances for each phase are j0.15 p.u.
and j0.05 p.u., respectively, and any mutual inductive coupling between phases
is neglected. Also, the zero-sequence impedances of both transformer and line
can be assumed to be equal to the corresponding positive- and negativesequence impedances.
• Calculate the fault current produced by a double-line-to-earth short
circuit, assuming that no load is connected to the line.
• How would the fault current change if the transformer primary circuit were
delta-connected? And if, on the contrary, the secondary circuit were deltaconnected?
Roberto Ferrero – ELEC 301/401 – 2018/2019
2
Step 1: Equivalent circuit
Thevenin equivalent circuit in the point of fault
• The network is operated at rated voltage:
𝐄 = 1 p. u.
• The equivalent impedance is the sum of
the transformer and line impedances:
𝐙𝑅 = 𝐙𝑌 = 𝐙𝐵 = 𝑗0.15 + 𝑗0.05 = 𝑗0.2 p. u.
In the original phase domain the circuit is described by the following equations:
𝐕𝑅
𝐙𝑅
𝐄
𝐕𝑌 = 𝐚2 𝐄 − 0
𝐕𝐵
0
𝐚𝐄
Roberto Ferrero – ELEC 301/401 – 2018/2019
0
𝐙𝑌
0
0
0
𝐙𝐵
𝐈𝑅
𝐈𝑌
𝐈𝐵
3
Step 2: Definition of fault
Mathematical constraints representing the fault
Definition of fault:
• VY = VB = 0
• IR = 0
0
1
𝐈𝑌 = 1
𝐈𝐵
1
1
𝐚2
𝐚
1
𝐚
𝐚2
𝐈0
𝐈1
𝐈2
𝐈0 + 𝐈1 + 𝐈2 = 0
𝐕𝑌 = 𝐚2 𝐄 − 𝐙𝑌 𝐈𝑌 = 𝐚2 𝐄 − 𝐚2 𝐙1 𝐈1 + 𝐚𝐙2 𝐈2 + 𝐙0 𝐈0 = 0
𝐕𝐵 = 𝐚𝐄 − 𝐙𝐵 𝐈𝐵 = 𝐚𝐄 − 𝐚𝐙1 𝐈1 + 𝐚2 𝐙2 𝐈2 + 𝐙0 𝐈0 = 0
𝐚 − 1 𝐙2 𝐈2 + 1 − 𝐚 𝐙0 𝐈0 = 0
Roberto Ferrero – ELEC 301/401 – 2018/2019
𝐙2 𝐈2 = 𝐙0 𝐈0
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Step 2: Definition of fault
Impedance matrix in the sequence domain
𝐙𝑅𝑌𝐵
𝐙𝑅
= 0
0
0
𝐙𝑌
0
𝑗0.2
0
0
0
𝑗0.2
0 p. u.
0 = 0
𝐙𝐵
0
0
𝑗0.2
• The impedance contributions arise from the transformer and line only:
𝐙1 = 𝐙2 = 𝐙𝑅 = 𝐙𝑌 = 𝐙𝐵 = 𝑗0.2 p. u.
• The transformer is grounded on both sides and the current path through the
ground has negligible resistance:
𝐙0 = 𝐙1 = 𝐙2 = 𝑗0.2 p. u.
𝐙0
𝐕0
0
𝐕1 = 𝐄 − 0
𝐕2
0
0
Roberto Ferrero – ELEC 301/401 – 2018/2019
0
𝐙1
0
0
0
𝐙2
𝐈0
𝐈1
𝐈2
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Step 3: Fault current calculation
Solution in the sequence domain and transformation in the original domain
The voltage source and the impedances in
the equivalent circuit are known:
𝐄 = 1 p. u.
𝐙0 = 𝐙1 = 𝐙2 = 𝑗0.2 p. u.
So the circuit can be solved:
𝐈1 =
𝐄
= −𝑗3.33 p. u.
𝐙1 + 𝐙2 𝐙0 / 𝐙2 +𝐙0
… and the fault current in the original domain can be calculated:
𝐙2
𝐈𝑓 = 𝐈𝑌 + 𝐈𝐵 = 2𝐈0 − 𝐈1 − 𝐈2 = 3𝐈0 = −3𝐈1
= 𝑗5 p. u.
𝐙2 +𝐙0
Roberto Ferrero – ELEC 301/401 – 2018/2019
6
Effects of the transformer connection
Delta-star connection
A zero-sequence current can flow in the secondary circuit of the transformer
because of its connection to earth and because a corresponding zerosequence current can flow within the delta in the primary circuit.
• The equivalent circuit representing the fault in the sequence domain is the
same as in the previous case:
 The fault current is unchanged:
𝐈𝑓 = 𝑗5 p. u.
Roberto Ferrero – ELEC 301/401 – 2018/2019
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Effects of the transformer connection
Star-delta connection (1)
In this case, zero-sequence currents can flow in the primary circuit and within
the delta of the secondary circuit, but not outside it, so not in the line.
• The zero-sequence circuit is open:
 The fault is now a line-to-line short
circuit, without any current flowing
through the ground.
Roberto Ferrero – ELEC 301/401 – 2018/2019
8
Effects of the transformer connection
Star-delta connection (2)
The circuit parameters are the same as in
the previous case, except for Z0:
𝐄 = 1 p. u.
𝐙1 = 𝐙2 = 𝑗0.2 p. u.
So the circuit can be solved:
𝐈1 =
𝐄
= −𝑗2.5 p. u.
𝐙1 + 𝐙2
… and the fault current in the original domain can be calculated:
𝐈𝑓 = 𝐈𝑌 = 𝐚2 − 𝐚 𝐈1 = −4.33 p. u.
Roberto Ferrero – ELEC 301/401 – 2018/2019
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