transformer3-perunit
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1
Per-Unit System
EE341 – Energy Conversion
Ali Keyhani
Transformer
Lecture #3
2
Per-Unit SystemIn the per-unit system, the voltages, currents, powers, impedances, and other electrical quantities are expressed on a per-unit basis by the equation:
Quantity per unit = Actual value
Base value of quantity
It is customary to select two base quantities to define a given per-unit system. The ones usually selected are voltage and power.
3
Per-Unit SystemAssume:
Then compute base values for currents and impedances:
ratedb VV
ratedb SS
b
bb V
SI
b
b
b
bb S
V
I
VZ
2
4
Per-Unit SystemAnd the per-unit system is:
b
actualup V
VV ..
b
actualup I
II ..
b
actualup S
SS ..
b
actualup Z
ZZ ..
%100% .. upZZ Percent of base Z
5
Example 1An electrical lamp is rated 120 volts, 500 watts. Compute the per-unit and percent impedance of the lamp. Give the p.u. equivalent circuit.
Solution:(1) Compute lamp resistance
power factor = 1.0
8.28500
)120( 222
P
VR
R
VP
08.28Z
6
Example 1(2) Select base quantities
(3) Compute base impedance
(4) The per-unit impedance is:
VASb 500
VVb 120
8.28500
)120( 22
b
bb S
VZ
..018.28
08.28.. up
Z
ZZ
bup
7
Example 1(5) Percent impedance:
(6) Per-unit equivalent circuit:
%100%Z
..01 upZ ..01 upVS
8
Example 2An electrical lamp is rated 120 volts, 500 watts. If the voltage applied across the lamp is twice the rated value, compute the current that flows through the lamp. Use the per-unit method.
Solution:
VVb 120
..02120
240.. up
V
VV
bup
..01.. upZ up
9
Example 2The per-unit equivalent circuit is as follows:
..01 upZ ..02 upVS
..0201
02
..
.... up
Z
VI
up
upup
AV
SI
b
bb 167.4
120
500
AIII bupactual 0334.8167.402..
10
Per-unit System for 1- CircuitsOne-phase circuits
LVbLV VV
IVSSb 1
whereneutraltolineVV
currentlineII
HVbHV VV
bLV
bbLV V
SI
bHV
bbHV V
SI
11
Per-unit System for 1- Circuits
b
bLV
bLV
bLVbLV S
V
I
VZ
2)(
b
bHV
bHV
bHVbHV S
V
I
VZ
2)(
*pupu
bpu IV
S
SS
cospupub
pu IVS
PP
sinpupub
pu IVS
12
Transformation Between BasesSelection 1
Ab VV 1Ab SS 1Then
11
b
Lpu Z
ZZ
1
21
1b
bb S
VZ
Selection 2
Bb VV 2Bb SS 2Then
22
b
Lpu Z
ZZ
2
22
2b
bb S
VZ
13
Transformation Between Bases
22
2
1
21
2
11
21
2
b
b
b
b
b
b
L
b
b
L
pu
pu
V
S
S
V
Z
Z
Z
Z
Z
Z
Z
Z
1
2
2
2
112
b
b
b
bpupu S
S
V
VZZ
“1” – old“2” - new
oldb
newb
newb
oldboldpunewpu S
S
V
VZZ
,
,
2
,
,,,
14
Transformation Between BasesGenerally per-unit values given to another base can be converted to new base by by the equations:
2
11___2___ ),,(),,(
base
basebaseonpubaseonpu S
SSQPSQP
2
11___2___
base
basebaseonpubaseonpu V
VVV
12
2
22
11___2___ )(
)(),,(),,(
basebase
basebasebaseonpubaseonpu SV
SVZXRZXR
When performing calculations in a power system, every per-unit value must be converted to the same base.
15
Per-unit System for 1- TransformerConsider the equivalent circuit of transformer referred to LV side and HV side shown below:
LVV HVV LVV HVV
SS jXR
1N 2N
22 a
Xj
a
R SS
(1) Referred to LV side (2) Referred to HV side
Define 12
1 N
N
V
Va
HV
LV
S
16
Per-unit System for 1- TransformerChoose:
ratedLVb VV ,1
ratedb SS
Compute:112
1bb
LV
HVb V
aV
V
VV
b
bb S
VZ
21
1 b
bb S
VZ
22
2
2
21
21
22
21
2
1
)1(
aV
a
V
V
V
Z
Z
b
b
b
b
b
b
Normal choose rated values as base values
17
Per-unit System for 1- TransformerPer-unit impedances are:
11..
b
SSup Z
jXRZ
121
22
2
22
2..b
SS
b
SS
b
SS
up Z
jXR
aZ
ajX
aR
ZajX
aR
Z
So:2..1.. upup ZZ
Per-unit equivalent circuits of transformer referred to LV side and HV side are identical !!
18
Per-unit Eq. Ckt for 1- Transformer
LVV HVV
SS jXR
1N 2N
Fig 1. Eq Ckt referred to LV side
12
1 N
N
V
Va
HV
LV
S
1bZ
1bV 2bV
Fig 2. Per-unit Eq Ckt referred to LV side Fig 3.
puSZ ,
1:1
1bV 2bVpuSZ ,
1bV 2bV
bS
19
Per-unit Eq. Ckt for 1- Transformer
LVV HVV
1N2N
Fig 4. Eq Ckt referred to HV side
12
1 N
N
V
Va
HV
LV
S
2bZ
2bV
Fig 5. Per-unit Eq Ckt referred to HV side Fig 6.
puSZ ,
1:1
1bV 2bVpuSZ ,
1bV 2bV
1bV
22 a
Xj
a
R SS
bS
20
Voltage Regulation
%100
loadfull
loadfullloadno
V
VVVR
Voltage regulation is defined as:
%100,
,,
loadfullpu
loadfullpuloadnopu
V
VVVR
In per-unit system:
Vfull-load: Desired load voltage at full load. It may be equal to, above, or below rated voltage
Vno-load: The no load voltage when the primary voltage is the desired voltage in order the secondary voltage be at its desired value at full load
21
Voltage Regulation ExampleA single-phase transformer rated 200-kVA, 200/400-V, and 10% short circuit reactance. Compute the VR when the transformer is fully loaded at unity PF and rated voltage 400-V.
Solution:
Fig 7. Per-unit equivalent circuit
PV SV1.0j
loadS
VVb 4002
kVASb 200
puS puload 01,
pujX puS 1.0,
SX
22
Voltage Regulation ExampleRated voltage:
puV puS 00.1,
pu
jj
XIVV
o
puSpupuSpuP
7.5001.1
1.011.000.100.1
,,,
puV
SI
puS
puloadpuload 00.1
00.1
00.1**
,
,,
23
Voltage Regulation Example
puVV opuPloadnopu 7.5001.1,,
puVV puSloadfullpu 00.1,,
Secondary side:
Voltage regulation:
%1.0%1000.1
0.1001.1
%100,
,,
loadfullpu
loadfullpuloadnopu
V
VVVR
24
Select Vbase in generator circuit and Sb=100MVA, compute p.u. equivalent circuit.
Problem 1
G
100j
20 kV 22kV/220kV80MVA14%
220kV/20kV50MVA10%
50MVA0.8 PF lagging
25
Per-unit System for 3- CircuitsThree-phase circuits
LVLbLV VV ,
IVSSSb 33 13 where 3/)(lineLneutraltoline VVV
Lcurrentline III
HVLbHV VV ,LLb IVS 3
bHVbHVbLVbLVb IVIVS 33
26
Per-unit System for 3- Circuits
b
bLV
b
bLVbLV
LV
LVbLV S
V
S
VV
I
VZ
2)(3
3
b
bHVbHV S
VZ
2)(
**
3
3
3pupu
bb
LL
bpu IV
IV
IV
S
SS
bLV
bbLV
V
SI
3
bHV
bbHV
V
SI
3
27
Three 25-kVA, 34500/277-V transformers connected in -Y. Short-circuit test on high voltage side:
Determine the per-unit equivalent circuit of the transformer.
Per-unit System for 3- Transformer
VV SCLine 2010,
AI SCLine 26.1,
WP SC 912,3
28
(a) Using Y-equivalent
Per-unit System for 3- Transformer
3
34500 277
SS jXR
VASb 250003
2010SCV
26.1SCI
00.92126.1
47.1160SCZ
VVSC 47.11603
2010
29
So
Per-unit System for 3- Transformer
86.90048.191921 2222
SSCS RZX
WP 3043
912 48.191
26.1
30422
SCS I
PR
86.90048.191 jZSC
VASb 25000 VV HVb 58.199183
34500,
99.1586925000
58.19918 2
,HVbZ
pujj
Z YpuSC 0568.0012.099.15869
86.90048.191,,
30
(b) Using -equivalent
Per-unit System for 3- Transformer
34500 277
,SCZ
VASb 250002010SCV
3
26.1SCI
79.2764727.0
2010,SCZ
VVSC 2010 AISC 727.03
26.1
31
So
Per-unit System for 3- Transformer
30.270418.57579.2764 222,
2
,, SSCS RZX
WP 3043
912 18.575
727.0
30422,
SCS I
PR
86.90048.191 jZSC
VASb 25000 VV HVb 34500,
4761025000
345002,HVbZ
pujj
Z puSC 0568.0012.047610
30.170418.575,,
32
Related Materials in Textbook(1) Section 2.6 and 2.7, page 83~90, Chapman
book(2) Section 2.10, page 113~116, Chapman book
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