PER-UNIT SYSTEM
In power systems many transformers at various (different) voltage levels are
involved. Per Unit System is a normalization procedure which provides a
mathematical basis for analyzing power networks with relative ease and
convenience. In addition, when various quantities are expressed in per unit
( pu ) or per cent values, they usually convey a message. For example, if a
bus voltage is 0.98 pu, it means that this value is 98 % of the nominal or
base value which could be at any level in the network. It also immediately
conveys a message that the value is an acceptable one. On the contrary, if
the voltage value is 1. 08, then it immediately conveys that the value is
higher than the acceptable level of 1.05 pu. Similar conclusions can be
drawn for other quantities such as current, power and impedance. The idea
here is to express various variables as a fraction of their corresponding base
(fixed) variables.
Quantity in per unit (p.u.) Actual Quantity
Base Value of Quantity (1)
There are several advantages offered by using per unit systems. These are
listed below.
SINGLE-PHASE SYSTEMS
The basic idea is to select two electrical variables such as power and voltage
as
independent base values. Then the base values for other two variables,
namely, current and impedance follow by Ohm’s Law. We will illustrate
this procedure for single-phase systems:
Let base value for power = S1B (single-phase)
base value for voltage = VB (line-neutral)
= VB (ln)
Then IB1 (line) = IB (l) = (2)
and ZB(y) = (3)
The ZB is on a per phase basis.
We will consider an example to illustrate the use of per unit system.
V Z = R + jX
One should realize that V, I, and Z are actual complex values. The base
values used for normalization are, however, real values. Therefore by
equation (1) the respective per unit values are also complex.
Now,
Vp.u. = (4)
Ip.u. = (5)
Zp.u. = (6)
= Rp.u. + jXpu (6a)
and Spu = (7)
or = (8)
= (9)
= Ppu + j Qpu (10)
Numerical Example
Let. V = 118 00 volts
Z = 5 300 ohms
Then I = 23.6 -300 amperes
& S = V I* = (118 00)(23.6 +300) va
= 2,784.8 300 va
For this example, it is appropriate to choose:
SlB = 3,000 va
VlB = 120-volts
Then IlB = = 25 amperes
& ZlB = = 4.8 ohms
Three-Phase Systems
Three-phase systems may be normalized by picking appropriate three-phase
bases. We will illustrate the various base choices for both Y and ∆ systems
on a comparative basis:
Wye (Y) Delta (∆ )
Choose (1) S3B = 3SlB (1) S3B = 3SlB
(2) VB (ll) = VB(ln) (2) VB(ll)
IB (l) = IB (l) =
Since S3B = VB(ll)IB(l)
IB(ph) =
ZB(Y) = ZB(∆ ) =
ZB(Y) = ZB(∆ ) =
Base Equations for ∆ Systems
Choose S3B as the three-phase apparent power base (rating if available) and
VB(ll) as the line-to line base. Here again if the system nominal line-to-line
is available or known, choose this value as the base.
IB (l) = or I1(pu) =
IB(ph) = ⇒ Iph(pu) =
ZB(∆ ) = ⇒ Z∆ (pu) =
Base Equations for Y Systems
Choose S3B as the three phase power base and VB(ll) as the line-to-line
voltage base.
Then IB(l) = =
where VB(ln) =
and ZB(Y) = =
This can also be shown =
Change of Base
It is often necessary to convert the base values of several pieces of
equipment connected together to form a power system (or interconnected
power system). Usually the name plate ratings of these individual devices
are different and hence their respective individual base values will be also
different. In order to refer all per unit values to a common system base, it is
necessary to change all device p.u. values to the common p.u. values. An
example later will illustrate this procedure. However, the key equation
development is as follows:
or = =
⇓⇓
l-l values three-phase values
Advantages of Per-Unit System (P.U.)
1. Per-unit representation results in a more meaningful and
correlated data. It gives relative magnitude information.
2. There will be less chance of missing up between single - and
three-phase powers or between line and phase voltages.
3. The p.u. system is very useful in simulating machine
systems on analog, digital, and hybrid computers for steady-
state and dynamic analysis.
4. Manufacturers usually specify the impedance of a piece of
apparatus in p.u. (or per cent) on the base of the name plate
rating of power ( ) and voltage ( ). Hence, it can be used
directly if the bases chosen are the same as the name plate
rating.
5. The p.u. value of the various apparatus lie in a narrow
range, though the actual values vary widely.
6. The p.u. equivalent impedance (Zsc) of any transformer is the
same referred to either primary or secondary side. For
complicated systems involving many transformers or different
turns ratio, this advantage is a significant one in that a possible
cause of serious mistakes is removed.
7. Though the type of transformer in 3-phase system,
determine the ratio of voltage bases, the p.u. impedance is the
same irrespective of the type of 3-phase transformer.
(Y ∆ , ∆ Y, ∆ ∆ , or Y Y)
8. Per-unit method allows the same basic arithmetic operation
resulting in per-phase end values, without having to worry
about the factor ‘100’ which occurs in per cent system.
Experience will, definitely, show the usefulness of the p.u.
system.
