Download Fault Analysis Symmetrical Components

Fault Analysis
Symmetrical Components
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Fault Analysis
• The cause of electric power system faults is
insulation breakdown
• This breakdown can be due to a variety of
different factors:
• Lightning.
• wires blowing together in the wind.
• animals or plants coming in contact with the
wires.
• salt spray or pollution on insulators.
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Fault Types
• There are two main types of faults
• Symmetric faults: system remains balanced; these faults
are relatively rare, but are the easiest to analyze so we’ll
consider them first.
• Un-symmetric faults: system is no longer balanced; very
common, but more difficult to analyze.
• The most common type of fault on a three phase system by
far is the single line-to-ground (SLG), followed by the line-toline faults (LL), double line-to-ground (DLG) faults, and
balanced three phase faults.
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Fault Analysis
• Fault currents cause equipment damage due to both
thermal and mechanical processes.
• Goal of fault analysis is to determine the magnitudes
of the currents present during the fault.
• need to determine the maximum current to insure
devices can survive the fault.
• need to determine the maximum current the circuit
breakers (CBs) need to interrupt to correctly size the
CBs.
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Fault Analysis Solution Techniques
•
Circuit models used during the fault allow the network to be
represented as a linear circuit
•
There are two main methods for solving for fault currents:
1.
Direct method: Use pre-fault conditions to solve for the
internal machine voltages; then apply fault and solve
directly.
2.
Superposition: Fault is represented by two opposing
voltage sources; solve system by superposition.
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Analysis of Un-Symmetric Systems
• Except for the balanced three-phase fault, faults result in
an unbalanced system.
• The most common types of faults are single line-ground
(SLG) and line-line (LL). Other types are double lineground (DLG), open conductor, and balanced three phase.
• System is only unbalanced at point of fault!
• The easiest method to analyze unbalanced system
operation due to faults is through the use of Symmetrical
Components
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Symmetric Components
• The key idea of symmetrical component analysis is to
decompose the system into three sequence networks.
The networks are then coupled only at the point of the
unbalance (i.e., the fault)
• The three sequence networks are known as the
• positive sequence (this is the one we’ve been using).
• negative sequence.
• zero sequence.
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Positive Sequence Sets
• The positive sequence sets
have three phase
currents/voltages with
equal magnitude, with
phase b lagging phase a by
120°, and phase c lagging
phase b by 120°.
• We’ve been studying
positive sequence sets.
Positive sequence sets
have zero neutral current
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Negative Sequence Sets
• The negative sequence sets
have three phase
currents/voltages with
equal magnitude, with
phase b leading phase a by
120°, and phase c leading
phase b by 120°.
• Negative sequence sets are
similar to positive
sequence, except the phase
order is reversed
Negative sequence sets
have zero neutral current
Zero Sequence Sets
• Zero sequence sets have three
values with equal magnitude and
angle.
• Zero sequence sets have neutral
current
Zero Sequence vectors with
Zero phase shift.
Sequence Set Representation
• Any arbitrary set of three phasors, say Ia, Ib, Ic can be
represented as a sum of the three sequence sets
I a  I a0  I a  I a
I b  I b0  I b  I b
I c  I c0  I c  I c
where
I a0 , I b0 , I c0 is the zero sequence set
I a , I b , I c is the positive sequence set
I a , I b , I c is the negative sequence set
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Conversion from Sequence to Phase
Only three of the sequence values are unique,
I0a , I a , I a ; the others are determined as follows:
 = 1120
0
0
0
Ia  I b  Ic
I b   2 I a
 2  3  0
3 1
(since by definition they are all equal)
I c   I a
I b   I a
I c   2 I a
0

1
1
1
1
I
 
 a
1 
 Ia 
1








0
+
2

2

 I   I 1  I   I   1 
   Ia 
b
a
a
a





 

 
 2  1   2   I  
 I c 
1
  
 
  a 
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Conversion Sequence to Phase
Define the symmetrical components transformation
matrix
1
1 1


2
A  1 

1   2 


0
0


I
I
a
 Ia 






Then I   I b   A  I a   A  I   A I s
 
 
 
 I c 
 I a 
 I 
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Conversion Phase to Sequence
By taking the inverse we can convert from the
phase values to the sequence values
1
Is  A I
1
1 1
1 1 
2
with A  1   
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1  2  


Sequence sets can be used with voltages as well
as with currents
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Example
Va   0 
Let V  Vb     
  

Vc     
Then
1   0   0 
1 1
1
1
2
Vs  A V  1         

 

3
1  2        6.12 


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Example
 I 0   100 
  
Let I s   I   10


     

 I  
Then
1   100    
1 1


2
I  AI s  1 
  10    

 

1   2       


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Power in Symmetrical Components
The total power in a three-phase network is given in
terms of phase variables by
where the asterisk denotes complex conjugation. We can show that the
corresponding expression in terms of sequence variables is given by
The total power is three times the sum of powers in individual
sequence networks.
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Use of Symmetrical Components
• Consider the following wye-connected load:
I n  I a  Ib  I c
Vag  I a Z y  I n Z n
Vag  ( ZY  Z n ) I a  Z n I b  Z n I c
Vbg  Z n I a  ( ZY  Z n ) I b  Z n I c
Vcg  Z n I a  Z n I b  ( ZY  Z n ) I c
Vag 
Z y  Zn
 

Vbg    Z n
V 
 Z
n
 cg 

Zn
Z y  Zn
Zn
  Ia 
 
Z n  Ib
 
Z y  Z n   I c 
Zn
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Use of Symmetrical Components
Vag 
Z y  Zn
Zn
 

Z y  Zn
Vbg    Z n
V 
 Z
Zn
n
 cg 

V
 Z I V  A Vs
A Vs  Z A I s

 Z y  3Z n

1
A ZA  
0

0

  Ia 
 
Z n  Ib
 
Z y  Z n   I c 
I  A Is
Zn
Vs  A
0
Zy
0
1
0

0
Z y 
Z A Is
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Networks are Now Decoupled
V 0 
 Z y  3Z n 0
 

0
Zy
V   
 

0
0
V
 

Systems are decoupled
V 0  ( Z y  3Z n ) I 0
0
0  I 
 
0  I 
 

Z y  I 
 
V
 Zy I
V   Zy I
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Grounding
• When studying unbalanced system operation how a
system is grounded can have a major impact on the fault
flows
• Ground current only impacts zero sequence system
• In previous example if load was ungrounded the zero
sequence network is (with Zn equal infinity):
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Grounding, cont’d
•
Voltages are always defined as a voltage difference.
The ground is used to establish the zero voltage
reference point
•
ground need not be the actual ground (e.g., an
airplane)
•
During balanced system operation we can ignore the
ground since there is no neutral current
•
There are two primary reasons for grounding electrical
systems
1. safety
2.
protect equipment
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Sequence diagrams for generators
• Key point: generators only produce positive sequence voltages; therefore
only the positive sequence has a voltage source.
During a fault Z+  Z  Xd”. The zero sequence impedance is
usually substantially smaller. The value of Zn depends on
whether the generator is grounded.
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Sequence diagrams for Transformers
• The positive and negative sequence diagrams for transformers are similar to those for
transmission lines.
• The zero sequence network depends upon both how the transformer is grounded and
its type of connection. The easiest to understand is a double grounded wye-wye
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Transformer Sequence Diagrams
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Unbalanced Fault Analysis
• The first step in the analysis of unbalanced faults is to assemble
the three sequence networks.
• For example, for the following power system let’s develop the
sequence networks.
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Sequence Diagrams
Positive Sequence Network
Negative Sequence Network
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Negative Sequence Network
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Zero Sequence Network
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