Department of Electronics and Electrical Engineering EE3 Power Systems

Department of Electronics
and Electrical Engineering

EE3 Power Systems
2. The Per-Unit System
2. The Per-unit System
In network or machine analysis usual to use per-unit
(pu) values
unit) same (in the value reference or base the
unit) any (in value actual the
unit per =
To a base of one kilowatt = !!! "
a #! " li$ht bulb rate% !.!# pu& a !! " bulb as !. pu.

! - #! " bulbs ' 2! - !! " bulbs
%issipate (!.!.!# ' 2!.!.) = 2.# pu power.
(eferrin$ back to the base power of !!! "
this represents a total power of 2#!! "
To a base power of !!! "& for a base volta$e of 2)! *&
base current woul% be !!!+2)! = , -.
.ach #! " bulb (!.!# pu power) takes !.!# pu current&
an% !!" bulb (!. pu power) takes !. pu current.
Total pu current is (!.!.!# ' 2!.!.) = 2.# pu to a base
of , - = !., -.
O = = = #2.)
,
2)!
current base
volta$e base
resistance /ase
0alculations of currents&
impe%ances& powers&
losses very cumbersome
- numerous volta$e levels
linke% by transformers.
"ithout the use of the per-
unit system
- convert power flows to
%ifferent volta$es an%
currents on either si%e of
transformers.
132 kV Busbar
Circuit breakers
Grid
delta-star, earthed
11 kV sitchb!ard
Circuit
Cable
Generat!r
neutral
earthed
"1# V sitchb!ard
$use sitch
$use and
c!ntact!r
Circuit breaker
Busbars
breakers
%u&iliary
Trans'!rmer
(!t!r
Generat!r
trans'!rmer
trans'!rmer
3.3 kV
).) kV
2. 1se in Power Systems -nalysis
2 This re%uces to
* ' a u l t + u
1.,+u
-!lta.e $ault
/'ault0+u1 2 1.,34'ault0+u1
132 kV Busbar
Circuit breakers
Grid
delta-star, earthed
11 kV sitchb!ard
Circuit
Cable
Generat!r
neutral
earthed
"1# V sitchb!ard
$use sitch
$use and
c!ntact!r
Circuit breaker
Busbars
breakers
%u&iliary
Trans'!rmer
(!t!r
Generat!r
trans'!rmer
trans'!rmer
3.3 kV
).) kV
"ith per-unit
transmission system becomes set of pu impe%ances
operatin$ at .! pu volta$e.
P1 impe%ances
2 e3hibit all the same relationships as absolute values&
2 obey all the same circuit laws& such as 4hms law an%
5irchhoffs 6aw&
7 but the arithmetic that is involved in the per-unit
calculations is much simpler.
"ithin three-phase power systems a per-unit phase
volta$e has the same numerical value as the
correspon%in$ per-unit line volta$e.
In the per-unit system three-phase values of volta$e&
current an% power not sensitive to \8 bein$ incorrect.
The actual values of resistance an% reactance for lines&
cables an% other apparatus are phase values.
The essence of per-unit calculations is
(a) 0hoose a base for all circuit parameters
9/ 7 one base may %efine others
(b) 0onvert all absolute values into per-unit
(c) 0arry out the very much simpler calculations
(%) 0onvert all per-unit values back into absolute terms
usin$ base in (a)
P1 analysis %epen%s on
2 consistent choice of a common base& an%
2 accurate conversion of circuit parameters into an% out
of that base.
:our base ;uantities re;uire% to completely %efine a per-
unit system
7 volta$e (*)&
7 current (I)&
7 comple3 power (S)&
7 an% impe%ance (<).
If two set arbitrarily& other two are automatically fi3e%
since S = * 3 I an% < = *+I
2.2 /ase =*-& *olta$e an% Impe%ance
In a sin$le phase system
Base apparent power (VA)
= Base current x Base voltage
= V
b
x I
b
Base MVA = kV
b
x kI
b
*
phase
I
phase
In a sin$le phase system
Base apparent power (VA)
= Base current x Base voltage
= V
b
x I
b
Base MVA = kV
b
x kI
b
In a three phase system
Base MVA per phase = kV
phase
x kI
phase
Base MVAtotal = 3. kV
phase
x kI
phase

= \3 kV
line
x kI
phase
*
phase
I
phase
*
phase
I
phase
*
line
If a current I
phase
flows in a circuit
un%er the action of volta$e *
phase

>>. it seems as if there is an
impe%ance somewhere in the
circuit limitin$ the current to that
value.
*
phase
I
phase
<
base
In power systems analysis I
phase
is %etermine% by
power transfer an% actual circuit impe%ance plays
no part in %etermination of <
base
*
base
I
base <
base
base
base
base
I
V

nt (A) Base curre
ge (V) Base volta
Z ance ( Base i!pe" = = )
=ultiply top an% bottom by *
base

an% prefi3 all ;uantities by !
8
*
base
I
base <
base
base
base
base
I
V

nt (A) Base curre
ge (V) Base volta
b
b
b b
b
b b
b
base
MVA
kV
x kI kV
kV
x I V
V
) ( Z
2 2 2
= = = O
=ultiply top an% bottom by *
base

an% prefi3 all ;uantities by !
8
*
base
I
base <
base
base
base
base
I
V

nt (A) Base curre
ge (V) Base volta
b
b
b b
b
b b
b
base
MVA
kV
x kI kV
kV
x I V
V
) ( Z
2 2 2
= = = O
:or three phase use *
line
an% total three phase =*-
base
) (
2
total MVA
kV
) ( Z
base
line
base
= O
2.2 Per-unit Impe%ance

ge (V) Base volta
tage (V) Actual vol
ge V unit volta #er
nt (A) Base curre
rent (A) Actual cur
nt I unit curre #er
) ance ( Base i!pe"
) e"ance ( Actual i!p
ance Z unit i!pe" #er
pu
pu
pu
=
=
O
O
=


ge (V) Base volta
tage (V) Actual vol
ge V unit volta #er
nt (A) Base curre
rent (A) Actual cur
nt I unit curre #er
) ance ( Base i!pe"
pu
pu
pu
=
=
O
O
=
*
phase
= 88!!+\8
I
phase
= ?)8-
*
line
= 8.8k*

ge (V) Base volta
tage (V) Actual vol
ge V unit volta #er
nt (A) Base curre
rent (A) Actual cur
nt I unit curre #er
) ance ( Base i!pe"
pu
pu
pu
=
=
O
O
=
*
phase
= 88!!+\8
I
phase
= ?)8-
*
line
= 8.8k*
pu
Z
Z
Z
phase
total MVA
kV
Z
MVA x x
I V total MVA
base
p
pu
base
line
base
line line base
!) . !
! . 2
. !
) (
) (
+phase !. con%uctor of impe%ance actual If
+ 2
,) . )
8 . 8
) (
,) . ) ?)2 ! 8 8 8
8 ) (
2 2
= =
O
O
=
O =
O = = =
= =
=
2.8 Per 1nit (eactances of Transformers
Transformer an% $enerator reactances (measurable)
may be ;uote% in absolute ohms or in per-unit.
2 Primary an% secon%ary leaka$e reactances are
summe% into one win%in$
2 Per-unit reactances are always ;uote%
to the base impe%ance of the %evice concerne%
*l +u
0onsi%er a ? =*- transformer& volta$e ratio
@8.8k*& with leaka$e reactance Al = !.!)pu
0alculate actual an% pu ;uantities in primary
!) . !
8 + !!!
8B
(phase) *base
volts -ctual
%rop Pu volts
8B* !.#B2 3 ,B2 (actual) %rop *olta$e
+ #B2 . ! ,, . 8 !) . !
+ ,, . 8
?
) (
+ ,B2
! 8
! ?
8
2 2
8
#
= = =
= =
O = = =
O = = =
= = =
phase x $ $ $
phase
total MVA
kV
$
phase A
x x
x
V
MVA
I
base pu pri!ar%
base
line
base
line
total
line
*l +u
(emember
0alculate actual an% pu ;uantities in secon%ary
!) . !
8 + 88!!
?)
(phase) *base
volts -ctual
%rop Pu volts
?)* !.!#!) )B)3 (actual) %rop *olta$e
+ !#!) . ! 2 . !) . !
+ 2 .
?
8 . 8
) (
+ )B)
! 8 . 8 8
! ?
8
2 2
8
#
= = =
= =
O = = =
O = = =
= = =
phase x $ $ $
phase
total MVA
kV
$
phase A
x x
x
V
MVA
I
base pu pri!ar%
base
line
base
line
total
line
*l +u
P1 volta$e %rop across transformer
= Ipu.Apu = .! . Apu Apu
irrespective of %istribution of leaka$e reactance
Same as
primary
2., P1 (eactances of Synchronous Cenerators
0onsi%er 8!="& #.#k*& !.Dpf
la$$in$ $enerator with A
s
= .2,pu
5t
V+
*s
/+
2, .
8 + ##!!
,B2,
(phase) *base
volts -ctual
%rop Pu volts
,B2,* .,, 82D!3 (actual) %rop *olta$e
+ ,, . #2 . 2, . ) (
+ #2 .
D . ! + 8!
# . #
) (
+ 82D!
! # . # 8
D . ! + ! 8!
8
2 2
8
#
= = =
= =
O = = = O
O = = =
= = =
phase x $ $ $
phase
total MVA
kV
$
phase A
x x
x
V
MVA
I
base pu s
base
line
base
line
total
line
9ote that =*-
base
= P(elec)+power factor = 8!="+!.D
2.) P1 (eactances of Synchronous =otors
5t
V+
*s
/+
0alculation of Apu for synchronous
motor is follows same process as
$enerator >>> but>>>
4utput power of a motor is always
;uote% as shaft mechanical power
an% P1 calculations use =*-
base
(elec)
=*-
base
= Pout(mech)
q . power factor
2.# 0onversions /etween /ases
In power systems analysis nee% to assert a common
=*-
base
an% convert all ohmic values into per unit
base% on operatin$ volta$e an% this =*-
base
If reactances or impe%ances are alrea%y in per-unit
to the base of the component (trans+$en+motor)
nee% to convert to same =*-
base
Since
2 2
). (
) ( ) (
line
base
base
line base
pu
kV
MVA
$
MVA
kV
$
$
$
$ O =
O
=
O
=
)
`
=
) (
) (
) ( ) (
ol" MVA
new MVA
ol" $ new $
base
base
pu pu
2.B 0onversions /etween off 9ominal *olta$es
In power systems analysis pu values of %evices
(trans+$en+motor) are ;uote% after calculation from
=*-
base
an% *
line
>> where *
line
is usually .!pu
4ccasionally transformers+motors+$enerators are
operate% at a volta$e off-nominal an% per-unit
calculations must allow for this.
Since
2 2
). (
) ( ) (
line
base
base
line base
pu
kV
MVA
$
MVA
kV
$
$
$
$ O =
O
=
O
=
)
`
=
) (
) (
) ( ) (
2
2
new kV
ol" kV
ol" $ new $
line
line
pu pu

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