Per-Unit System
In power systems there are so many different
elements such as Motors, Generators and
Transformers with very different sizes and
nominal values.
To be able to compare the performances of a
big and a small element, per unit system is
used.
Power System Representation
Physical Components in the system arerepresented by a mathematical model.
Mathematical models of components are
connected in exactly the same way as the physical components to obtain the system representation.
Various physical components have different
ratings or basis.
It is convenient to obtain the representation with
Per Unit And Percent Representation
The numerical per unit value of any
quantity is its ratio to the chosen base quantity of the same dimensions.
Thus a per unit quantity is a normalized
quantity with respect to a chosen base value.
Percent is the per unit quantity multiplied
Per Unit And Per Cent Representation Advantages
In the per-unit system of representation, device
parameters tend to fall in a relatively fixed range, making erroneous values prominent.
Ideal transformers are eliminated as circuit
elements. This results in a large saving in component representation and reduces computational burden.
The voltage magnitude throughout a given power
system is relatively close to unity in the per-unit system for a power system operating normally. This characteristic provides a useful check on the calculations.
Per Unit And Percent Representation Base Quantities
In power system calculations the nominal
voltage of lines and equipment is almost always known, so the voltage is a
convenient base value to choose.
The apparent power (volt-ampere) is
usually chosen as a second base. In
equipment this quantity is usually known and makes a convenient base.
Per Unit And Percent Representation Base Quantities
The choice of these two base quantities will
automatically fix the base of current, impedance, and admittance.
In a system study, the volt-ampere base can
be selected to be any convenient value such as 100 MVA, 200 MVA, etc.
Per Unit And Percent Representation Base Quantities
The same volt-ampere base is used in all
parts of the system. One base voltage in
a certain part of the system is selected
arbitrarily. All other base voltages must
be related to the arbitrarily selected one
by the turns ratio of the connecting
Per Unit System
Power system quantities such as voltage,
current and impedance are often expressed in per unit or percent of specified values.
Per unit quantities are calculated as:
Value
Base
Value
Actual
Value
Per Unit
Per Unit Values base pu S S S base pu I I I base pu V V V base pu Z Z Z Z Z 2 base base base pu V S Z Z pu base 2 base pu base Z S V Z Z Z
Conversion of Per Unit Values
Per Unit System
Usually, the nominal apparent power (S) and
nominal voltage (V) are taken as the base
values for power (S
base) and voltage (V
base).
The base values for the current (I
base) and
impedance (Z
base) can be calculated based on
the first two base values.
Per Unit System
100%
Z
base actual %
Z
Z
The percent impedance
e.g. in a synchronous generator with 13.8 kV
as its nominal voltage, instead of saying the voltage is 12.42 kV, we say the voltage is 0.9 p.u.
Per Unit in 3
Circuit
Simplified:
Concerns about using phase or line voltages are
removed in the per-unit system
Actual values of R, XC and XL for lines, cables,
and other electrical equipment typically phase values.
It is convenient to work in terms of base VA (base
Per Unit in 3
Circuit
,
3
3
B B B B B BV
S
I
I
V
S
B B B B B B B BS
V
I
V
Z
Z
I
V
23
/
3
• Usually, the 3-phase SB or MVAB and line-to-line VB or kVB are selected
Change of Base
The impedance of individual generators &
transformer, are generally in terms of
percent/per unit based on their own ratings.
Impedance of transmission line in ohmic value When pieces of equipment with various
different ratings are connected to a system, it is necessary to convert their impedances to a per unit value expressed on the same base.
Z V S Z Z Z old B old B old B old pu 2
Z V S Z Z Z new B new B new B new pu 2 new B new B V base voltage new & S base power new on the impedance unit per new the be new pu Z old B old BV
base
voltage
&
S
base
power
on the
impedance
unit
per
the
be
old puZ
1 2Change of Base
2
new B old B old B new B old pu new puV
V
S
S
Z
Z
old B new B old pu new puS
S
Z
Z
From (1) and (2), the relationship between the old and the new per unit value
If the voltage base are the same,
General Relations Between Circuit
Quantities
3 2 3 3 3 2 3 30 3 3 * 30 3 30 3 3 30 3 30 30 3 * 3 30 S V S V V I V I V Z V S Z V I I Connection S V S V V I V Z Connection Y o LL LL o LL L o LL D LL D LL o D LL o L D o LL LL o LL L LN Y
Base Conversion
2 ) ( 2 ) ( ) ( ) ( ) ( ) (
*
*
new base old base old base new base old pu new puKV
KV
MVA
MVA
Z
Z
Example One: Base Conversion
• A 50-MVA, 34.5:161 kV transformer with 10%
reactance is connected to a power system where all the other impedance values are on a 100 MVA, 34.5 or 161 kV base. The reactance of the transformer under new base is:
2
.
0
*
50
100
*
1
.
0
2 ) ( 2 ) ( ) (
new base old base new puKV
KV
Z
Example Two: Base Conversion
• A generator and transformer, as shown below, are to
be combined into a single equivalent reactance on a 100 MVA, 110 kV (high voltage side) base.
Advantages
• More meaningful when comparing different voltage
levels
• The per unit equivalent impedance of the transformer
remains the same when referred to either the primary or the secondary side
• The per unit impedance of a transformer in a
three-phase system is the same, regardless the winding connection
• The per unit method is independent of voltage
Advantages
• Manufacturers usually specify the impedance of the
equipment in per unit or percent on the base of its nameplate ratings
• The per unit impedance values of various ratings of
Advantages
Transformer equivalent circuit can be
simplified by properly specifying base
quantities.
Give a clear idea of relative magnitudes of
various quantities such as voltage, current, power and impedance.
Avoid possibility of making serious calculation
error when referring quantities from one side of transformer to the other.
Advantages
Per-unit impedances of electrical equipment of
similar type usually lie within a narrow
numerical range when the equipment ratings
are used as base values.
Manufacturers usually specify the impedances of
machines and transformers in per-unit or percent in nameplate rating.
Advantages
The circuit laws are valid in per unit systems,
and the power and voltage equation are
simplified since the factor √3 and 3 are
eliminated in the per-unit systems.