Symmetrical Components
1

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.
2

• 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. 3

• 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. 4

• 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.
2.
Direct method: Use pre-fault conditions to solve for the internal machine voltages; then apply fault and solve directly.
Superposition: Fault is represented by two opposing voltage sources; solve system by superposition.
5

• 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
6

• 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.
7

• 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
8

• 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 have three values with equal magnitude and angle.
• Zero sequence sets have neutral current
Zero Sequence vectors with
Zero phase shift.

• Any arbitrary set of three phasors, say I a
, I b
, I c
can be represented as a sum of the three sequence sets
I
I a b
I c where

I
0 a

I
 a

I
 a

I
0 b

I b
 
I b


I c
0 
I c
 
I c

I
0 a
, I
0 b
, I c
0
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
11

Conversion from Sequence to Phase
Only three of the sequence values are unique,
I , I
 a
, I
 a
; the others are determined as follows:

= 1 120
   2   3 
0
 3 
1
I
0 a

I
0 b

I (since by definition they are all equal)
I b
   2
I
 a
I c
  
I
 a
I b
  
I
 a
I c
   2
I
 a
 
 
 
I b

I
0 a
1
1
1

I
+ a



I
 a


1 1 1



1
1
 2 
  2




I
I

 
I

12

Conversion Sequence to Phase
Define the symmetrical components transformation matrix
A
Then
 1 1 1


1
1
 2 
  2




I

I
 
I

A
 
 
 
 
I
 
 

A  
 
 

A I s
13

Conversion Phase to Sequence
By taking the inverse we can convert from the phase values to the sequence values
I s
 with A

1

1 1 1

1
3
 1

 
2 
2




Sequence sets can be used with voltages as well as with currents
14

Let V

V a
  
V
V b c
  




Then
V s

A V

1
3
 1 1 1



1
1
 
 2 
2
 
 
 
6.12





15

Let I s

I
  
I







Then
I

AI s


1 1 1



1
1
 2
 

2

10 0
 
 
 
 

    




16

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.
17

Use of Symmetrical Components
• Consider the following wye-connected load:
I n

I a

I b

I c
V ag

I Z y

I Z n
V ag

( Z
Y

Z n
) I a

Z I

Z I
V bg

Z I

( Z
Y

Z n
) I b

Z I
V cg

Z I

Z I

( Z
Y

Z n
) I c



V
V ag bg








 V cg




Z y

Z n
Z
Z n n
Z
Z n
Z y

Z n n
Z
Z n n

I b
Z y

Z n

I
18

Use of Symmetrical Components



V
V ag bg








V cg
 
Z y

Z n
Z
Z n n
V
A V s


Z y
Z
 n
Z n
Z
Z n n

I
Z n
Z I V

A V s
Z A I s

Z y

Z n

I
I

A I s
V s

A

1
Z A I s b
A

1
Z A
 



Z y

3 Z n
0
0
Z
0
0 y
0
0
Z y




19

Networks are Now Decoupled

V
V
0


 
 

 V








Z y

3 Z
0
0 n
Systems are decoupled
V
0 
V
 
( Z y


3 Z n
) I
0
Z
0
0 y
0
0
Z y




I
 
I
 
I

V
  
20

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 Z n equal infinity):
21

• 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.
2.
safety protect equipment
22

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.
23

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
24

Transformer Sequence Diagrams
25

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.
26

Sequence Diagrams
Positive Sequence Network
Negative Sequence Network
27

Negative Sequence Network
28

Zero Sequence Network
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