4 Symmetrical Fault Analysis

4 Symmetrical Fault Analysis

Shital Patel, EE Department Power System - II (3150911) 1

4.1. Types of electrical fault Electrical fault is the deviation of voltages and currents from nominal values or states.

Under normal operating conditions, power system equipment or lines carry normal

voltages and currents which results in a safer operation of the system. But when fault

occurs, it causes excessively high currents to flow which causes the damage to

equipments and devices.

Causes of electrical faults are

o Weather conditions

Lightning strikes

Heavy rains

Heavy winds

Salt deposition

Snow and ice accumulation

o Equipment failures

Malfunctioning

Ageing

Insulation failure of cables and winding

o Human errors

Improper rating of equipment or devices

Forgetting metallic or electrical conducting parts after servicing or maintenance

Switching the circuit while it is under servicing

o Smoke of fires

Ionization of air, due to smoke particles, surrounding the overhead lines results in

spark between the lines or between conductors to insulator

Electrical faults are classified as

o Short circuit faults: A short circuit fault is an abnormal connection of very low

impedance between two points of different potential. These are the most common and

severe kind of faults, resulting in the flow of abnormal high currents through the

equipment or transmission lines. Short circuit faults are also called as shunt faults.

Symmetrical faults: These are very severe faults and occur infrequently in the

power systems. It affects each of the three phases equally. In transmission line

faults, roughly 5% are symmetric.

Line to line to line to ground (L-L-L-G)

Line to line to line (L-L-L)

Unsymmetrical faults: These are very common and less severe than symmetrical

faults. It does not affect each of the three phases equally. In transmission line

faults, roughly 95% are unsymmetrical.

Line to ground (L-G)



Line to line (L-L)

Line to line to ground (L-L-G)

o Open circuit faults: These faults occur due to the failure of one or more conductors.

Open circuit faults are also called as series faults. These are unsymmetrical or

unbalanced type of faults except three phase open fault.

One conductor open

Two conductor open

o Transient fault: A transient fault is a fault that is no longer present if power is

disconnected for a short time and then restored. Basically, transients are momentary

changes in voltage or current that occur over a short period of time i.e. interval is

usually described as approximately one sixteenth of a voltage cycle.

Electrical faults on power system has very ill effect such as

o Abnormal operation of the system

o Danger fatal accidents

o Insulation failures of the equipment

o Fire and explosion

o Equipment overheating

o Low system voltages

o Restricted power flow

o Complete blackout

4.2. Transient on a transmission line

In power system, selection of switchgear not only depends on rated current but it also

depends on momentary maximum current that supposed to flow under abnormal or

short circuit condition.

Hence it becomes important to find out initial maximum current at the time of short

circuit.

Consider a line fed from constant voltage source and short circuit takes place when line

is unloaded.

For simplification line capacitance is neglected hence analysis of the initial maximum

current is the study of transients in R-L series circuit.

Let, a series R-L circuit is connected to the source at the instant t=0.

( )

( )

m

m

div Ri L

dt

diV Sin t Ri L

dt

Vdi Ri Sin t

dt L L

The corresponding characteristic equation is 0R

sL

and the solution of this equation



has two parts, complementary function and particular integral.

The complementary function is,Rt

Lci ke

The particular integral is, ( ) ( )pi ACos t BSin t

The constants A and B undefined and can be found out by putting the value of ip in original

equation.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

m

m

m

dV Sin t R ACos t BSin t L ACos t BSin t

dt

V Sin t RACos t RBSin t LASin t LBCos t

V Sin t RB LA Sin t RA LB Cos t

Comparing the equation on both the side

& 0mV RB LA RA LB

Solving both the equation, value of A and B can be

2 2 2 2 2 2&m mLV RV

A BR L R L

Substituting the value of A and B in the equation

2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

2 2 2

( ) ( )

( ) ( )

( ) ( )

( )

m mp

m

m

m

LV RVi Cos t Sin t

R L R L

V L RCos t Sin t

R L R L R L

VSin t Cos Cos t Sin

R L

VSin t

R L

Hence total solution is

2 2 2( )

Rt

mLc p

Vi i i ke Sin t

R L

The constant k can be evaluated by knowing the initial condition. When switch is closed

at t=0, the initial current in inductor is zero i.e. i(0-) = i(0)=i(0+)=0.

0

2 2 2

2 2 2

0 ( 0 )

( )

R

mL

m

Vke Sin

R L

Vk Sin

R L

By putting the value of k, the solution of R-L series circuit is



2 2 2 2 2 2

2 2 2 2 2 2

2 2 2

( ) ( )

( ) ( )

( ) ( )

Rt

m mL

Rt

m m L

Rt

m L

V Vi Sin e Sin t

R L R L

V VSin t Sin e

R L R L

VSin t Sin e

R L

In above equation has two responses

( ) Steady state response

( ) Transient responseRt

L

Sin t

Sin e

Sinusoidal steady state current is called symmetrical short circuit current and

unidirectional transient current is called DC off set current.

2 2 2

2 2 2

( )

( )

ms

Rt

m Lt

Vi Sin t

R L

Vi Sin e

R L

The maximum momentary short circuit current at the instant of first peak (decay of

transient current in this short time is neglected and transmission line resistance is very

small i.e. =90°) is

2 2 2 2 2 2

m mmm

V Vi Cos

R L R L

The maximum possible value of momentary short circuit current occurs when voltage

wave is passing through zero i.e. α=0.

max 2 2 2 2 2 2

2 2 2

2 2 2

0

2

22

m mmm

m

V Vi Cos

R L R L

V

R L

V

R L

Above equation states that maximum possible momentary short circuit current value is

twice the maximum of symmetrical short circuit current i.e. doubling effect.

Usually selection of switchgear ratings are carried out on the bases of maximum possible

momentary short circuit current as safe choice.



Vis

t

t

it

i = is + it

t

Maximum Momentary Current, imm

Modern day switchgears are designed to interrupt the current in first few cycle it means

when short circuit current is interrupted, DC off set current has not deceased and so it

contributes to the current to be interrupted.

For the contribution, amount of DC off set current is not calculated but symmetrical short

circuit current alone is calculated and then the value is increased by multiplying factor to

account for DC off set current.

Switchgear operation speed Multiplying factor

8 cycles or slower 1.0

5 cycles 1.1

3 cycles 1.2

2 cycles 1.4



4.3. Short circuit on synchronous machine when unloaded Under steady state short circuit condition, the armature reaction of a synchronous

generator produces a demagnetizing flux and this effect is modelled as a reactance Xa in

series with the induced emf.

The total reactance leakage reactance Xl and armature reactance Xa of the machine is

called synchronous reactance Xd.

When sudden three phase short circuit occurs at the unloaded synchronous generator i.e.

open circuit condition, it undergoes a transient in all three phases by gradually reaching

to steady state condition.

The circuit breaker has to interrupt the short circuit current before steady state

condition.

When fault occurs, immediately a DC off set current appears in all three phase, each with

different magnitude because each voltage wave has different magnitude at the time of

occurrence of fault.

At the instant of short circuit, current is limited by leakage reactance of machine. Since

airgap flux cannot change instantaneously to counter the demagnetization of armature

short circuit current, current appears in the field winding as well as in damper winding

in the direction of main flux.

This current decays depending upon the time constants of winding. Time constant of

damper wing is much less than the field winding, thus during initial short circuit, damper

winding reactance and field winding reactance appears in parallel with armature

reactance.

During middle period of short circuit damper winding current dies out and it becomes

open circuit. At this instant only field winding reactance appears in parallel with

armature reactance.

During finally steady state period of short circuit, field winding current dies out and it

also becomes open circuit. At this instant only armature reactance appears.

o The reactance of synchronous machine appeared in initial short circuit is called sub

transient reactance (Xd’’).

o The reactance of synchronous machine appeared in middle of short circuit is called

transient reactance (Xd’).

o The reactance of synchronous machine appeared in end of short circuit is called

synchronous reactance (Xd).



Xf

+

Eg

Xl

Direct axis subtransient reactance

Xdw

Xa

+

Eg

Xl

Direct axis transient reactance

Xf

Xa

Xa

+

Eg

Xl

Xd

Synchronous Reactance

" "

"

' '

'

1

1 1 1

1

1 1

1

1

g

d l

d

a f dw

g

d l

d

a f

g

d l l a

d

a

EX X I

XX X X

EX X I

XX X

EX X X X I

XX

Synchronous machine is offering time varying reactance from Xd” to Xd’ and finally Xd such

that Xd” < Xd’ < Xd.

4.4. Short circuit on synchronous machine when loaded Analysis of short circuit on synchronous machine when loaded is too complicated, hence

several methods are used to carry out short circuit analysis.

Let, synchronous generator operating under steady conditions.

0

0

"

'

Load current

Bus terminal voltage

Induced emf

Direct axis synchronous reactance

Voltage behind subtransient reactance

Voltage behind transient reactance

g

d

g

g

I

V

E

X

E

E

When short circuit occurs at the synchronous generator terminal, induced emf can be

modelled as

" 0 0 "

' 0 0 '

g d

g d

E V jI X

E V jI X

For synchronous motor internal emf and reactances are similar to that of synchronous

generator except that the direction of current is reversed.

" 0 0 "

' 0 0 '

m d

m d

E V jI X

E V jI X



In case of short circuit of an interconnected system, the synchronous machines i.e.

generators and motors are replaced by their corresponding circuit models having voltage

behind sub transient/ transient reactance in series with sub transient/transient

reactance. The rest of the network being passive remains unchanged.

4.5. Short circuit current calculation using Thevenin Theorem

Analysis of short circuit on synchronous machine through Thevenin theorem is faster and

easy for large network.

Consider a synchronous generator feeding a synchronous motor through transmission

line under steady load condition. A symmetrical fault is made at F, at the motor terminals.

For fault analysis, synchronous machines are represented by their sub transient

reactances or transient reactances if transient currents are of interest in series with

voltages behind sub transient reactances. This change does not disturb the prefault

current I0 and prefault voltage V0 at F.

+

I0

+

X

V0

F

G

+

I0

+

X

V0

F

G

Xdg

Eg

Xdm

Em

X dg

E g

X dm

E m

Thevenin equivalent circuit looking from FG

XF

G

V0

XF

G

V0+

X dg X dm X dg X dm

DIg DIm

If

Zf

In a case of fault at F through an impedance Zf.

0 0

'' ''

'' ''

f

fTH dm dg f

dm dg

V VI

jX Z X X Xj Z

X X X



Current through generator and motor circuit

''

'' ''

''

'' ''

fdmg

dg dm

dg fm

dg dm

XI I

X X X

X XI I

X X X

D

D

Postfault currents and voltages

'' 0

'' 0

0 0

g g

m m

f fTH

I I I

I I I

V V jX I V V

D

D

D

So, the prefault current flowing out of the fault point F is always zero and the postfault

current out of F is independent of load for a given prefault voltage at F.

Steps for solving short circuit current by thevenin theorem approach

o Obtain the steady state solution using load flow and formulate the circuit model.

o Replace reactance of synchronous machine by their transient or sub transient values.

o Short Circuit all emf sources and find the value of or TH THZ X .

o Compute the required currents using Thevenin’s theorem.

Assumptions

o All prefault voltage equal to 1pu.

o Load current neglected as it is very less as compared to short circuit current.

4.6. Criteria for selection of circuit breaker Usually selection of switchgear ratings are carried out on the bases of maximum possible

momentary short circuit current as safe choice.

Modern day switchgears are designed to interrupt the current in first few cycle it means

when short circuit current is interrupted, DC off set current has not deceased and so it

contributes to the current to be interrupted.

For the contribution, amount of DC off set current is not calculated but symmetrical short

circuit current alone is calculated and then the value is increased by multiplying factor to

account for DC off set current.

Switchgear operation speed Multiplying factor

8 cycles or slower 1.0

5 cycles 1.1

3 cycles 1.2

2 cycles 1.4

If short circuit MVA is greater than 500, than above value is increase by 0.1 each.

Current that a circuit breaker interrupts is inversely proportional to the operating voltage



over a certain range. Of course, operating voltage is not allowed to exceed maximum

design value.

No matter how voltage is low, rated interrupting current should not exceed rated

maximum interrupting current i.e. over this range of voltage, product of operating voltage

and interrupting current is constant.

Rated voltageAmperes at operating voltage = Ampere at rated voltage ×

Operating voltage

line linerated rated interrupting currentRated interrupting MVA = 3 V I

Short circuit MVA = 3 × Pre fault line voltage (kV) × Short circuit current (kA)

For the selection of a circuit breaker, a maximum possible short circuit MVA to be

interrupted with respect to type and location of fault and generating capacity connected

to the system is required to calculate.

A three phase fault i.e. very rare is generally the one that has highest short circuit MVA

and a circuit breaker must be capable to interrupt it.

4.7. Algorithm for short circuit studies It is always necessary to develop a systematic approach to carry out short circuit

calculation of large system.

Consider, n-bus system operating under steady load. It is required to first calculate,

prefault voltages at all buses and current through line and rth bus gets faulty through fault

impedance Zf and hence the voltage at each bust is about to change.

G2G1 Gr Gn

System

1

2

n

r

System

1

2

n

r

X dnX drX d2X d1

V0r

+

-

Zf

If

0 Prefault bus voltages vector

Postfault bus voltages vector

V Changes in bus voltages vector due to fault

Bus impedance matrix of Thevenin's network

Bus current injection vector

Faul

f

f

BUS

fBUS

BUS

V

V

Z

J

Z

D

0

t impedance vector

Prefault current in line vector

Postfault current in line vector

BUS

fBUS

I

I



Step-1: Calculate prefault voltages and postfault bus voltages at all buses

01

00 2

0

0

BUS

n

fBUS BUS

V

VV

V

V V V

D

Step-2: Draw Thevenin’s equivalent circuit i.e. short circuit all emf sources and replace all

reactances by their respective transient reactances transient or sub transient reactances.

Step-3: Calculate DV

1

211 1

1

11 1

1

0

0

0

f

BUS

f

f

n

fr

n nn

fn

n

fr

n nn

frr

V Z J

I

IZ Z

I

Z Z

I

Z Z

V I

Z Z

Z I

D

D

Step-4: Calculate the postfault voltage for the rth bus

0 0

0

0

0

fr r r

fr rr

f f fr rr

f rf

rr

V V V

V Z I

Z I V Z I

VI

Z Z

D

Step-5: Calculate the postfault voltage for any ith bus

0 0

0

00

0 0

fi i i

fi ir

ri ir f

rr

iri rf

rr

V V V

V Z I

VV Z

Z Z

ZV V

Z Z

D

For i = r, post fault voltage at ith bus



0 0

0 0

0

0

f irr i rf

rr

rrr rf

rr

frr rr

rfrr

f

rfrr

ZV V V

Z Z

ZV V

Z Z

Z Z ZV

Z Z

ZV

Z Z

Step-6: Find the post fault current in lines

f f fij ij i jI Y V V

4.8. ZBUS formation by step by step method It is a step by step modelling technique that proceeds branch by branch. It is more

advantages because any modification in the network does not require complete

reconstruction of ZBUS

Let us consider that a ZBUS of a certain system is already constructed and at later stage

another branch is added.

Due to addition of a new branch, below mentioned situation can be possible.

Type – 1 modification: Branch is added between new bus and reference bus

It is the modification where Zb is added from a new bus to the reference bus i.e. a new

branch is added and dimension of ZBUS goes up by one.

Consider a passive n-bus network in which branch with impedance Zb is added to the new

bus k and reference bus r.

Passive Linear n-bus

network

1

n

i

j

k

rVk

Ik

Zb

0; 1,2,3, ,

k k k

ki ik

kk b

V Z I

Z Z i n

Z Z

0

0 0

BUS

BUS

b

Z old

Z new

Z

Type – 2 modification: Branch is added between new bus and old bus

It is the modification where Zb is added from a new bus to an old bus i.e. a new branch is

added and dimension of ZBUS goes up by one.

Consider a passive n-bus network in which branch with impedance Zb is added from new

bus k to old bus j.




network

1

n

i

j

k

r

Ij

Ik

Vk

Ij+Ik

1 1 2 2

1 1 2 2

k b k j

b k j j jj j k jn n

j j jj j jn n b jj k

V Z I V

Z I Z I Z I Z I I Z I

Z I Z I Z I Z I Z Z I

1

2

1 2

j

BUS j

BUS

jn

j j jn jj b

Z

Z old Z

Z new

Z

Z Z Z Z Z

Type – 3 modification: Branch is added between old bus and reference bus

It is the modification where Zb is added from old bus to reference bus i.e. a new loop is

formed but dimension of ZBUS does not change.

Consider a passive n-bus network in which branch with impedance Zb is added from old

bus j to reference bus r by setting Vk = 0


network

1

n

i

j

r

Zb

1 1 1

2 2 2

1 20

j

BUS j

n jn n

j j jn jj b k

V Z I

V Z old Z I

V Z I

Z Z Z Z Z I

1 1 2 2

1 1 2 2

1 1 2 2

0

1

k b k

b k j j jj j k jn n

j j jj j jn n b jj k

k j j jj j jn n

b jj

V Z I

Z I Z I Z I Z I I Z I

Z I Z I Z I Z I Z Z I

I Z I Z I Z I Z IZ Z

1 1 2 2

1 2

1 1 2 2

i i i in n ij j

ij j ij j ij jn

i i in n

b jj b jj b jj

V Z I Z I Z I Z I

Z Z Z Z Z ZZ I Z I Z I

Z Z Z Z Z Z

1

2

1 2

1

j

j

BUS BUS j j jn

jj b

jn

Z

ZZ new Z old Z Z Z

Z Z

Z

Type – 4 modification: Branch is added between two old buses

It is the modification where Zb connects two old bus i.e. a new loop is formed but

dimension of ZBUS does not change.



Consider a passive n-bus network in which branch with impedance Zb connects two old

buses.


network

1

n

i

j

r

Ii+Ik

Ij-Ik

Zb

Ik

1 1 2 2

1 1 2 2

1 1 2 2

1 1 1 2 2 2

...

0

i i i ii i k ij j k in n

j b k i

j j ji i k jj j k jn n

b k i i ii i k ij j k in n

i j i j ii ji i ij jj j in jn n b i

V Z I Z I Z I I Z I I Z I

V Z I V

Z I Z I Z I I Z I I Z I

Z I Z I Z I Z I I Z I I Z I

Z Z I Z Z I Z Z I Z Z I Z Z I Z Z

i jj ij ji kZ Z Z I

11 1 1

22 2 2

1 1 2 20 2

i j

BUS i j

nn in jn

i j i j jn jn b ii jj ij j

IV Z Z

IV Z old Z Z

IV Z Z

Z Z Z Z Z Z Z Z Z Z I

1 1

2 2

1 1 2 2

1

2

i j

i j

BUS BUS i j i j in jn

b ii jj ij

in jn

Z Z

Z ZZ new Z old Z Z Z Z Z Z

Z Z Z Z

Z Z

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4 Symmetrical Fault Analysis