Page 90 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 79

When quantities are expressed in per-unit, most voltages are close to one under
normal operation; higher values indicate overvoltage and lower values may indicate
overload. This helps the engineer in scanning the results of a load-flow analysis
or fault study, because abnormal conditions are immediately recognizable: for exam-
ple, currents outside the 0±1 range, or voltages that deviate more than a few per cent
from one. Under transient conditions, larger voltage and current swings may be
encountered.
p
Engineering formulas often contain `funny' coefficients such as 2p/2 or even
constants such as 1.358, and it is often far from obvious where these came from. 18 In
a well-chosen per-unit system, factors that are common to both the actual and the
base values cancel out, and this gets rid of many of the spurious coefficients. Per-unit
expressions are therefore less cluttered and express the essential physical nature of the
system economically.
An example of this is the normalization of power in a three-phase system: in
ordinary units
p 
P ˆ 3V LL I L cos f [W] (2:71)
The bases P b , V b and I b must be related by the same equation: thus
p 
P b ˆ 3V b I b ( cos f) b [W] (2:72)
Normalizing means dividing equation (2.71) by equation (2.72). Taking the base
value of the power factor to be (cos f) ˆ 1, we get
b
p ˆ vi cos f (2:73)
p
The factor 3 cancels out: this not only simplifies computation, but also expresses
the power equation in a fundamental general form that is independent of the number
of phases or whether the load is connected in wye or delta.

2.13.1 Standard formulas for three-phase systems

A three-phase system is rated according to its MVA capacity, S. Let the base value
for volt-amperes be S b MVA. If the base line±line voltage is V b and the base line
current is I b , then
p 
S b ˆ 3V b I b (2:74)

Usually, V b is expressed in kV and I b is in kA. By convention the base impedance Z b
is the line-neutral impedance
p 
V b = 3
Z b ˆ
(2:75)
I b

18
They usually arise in the theoretical derivation of the formula but they may arise because of the parti-
cular units used for parameters in the equation. A simple example is the equation y ˆ 25:4x to represent a
length y in mm that is equal to another length x in inches. The equation expresses the essential equality of
the dimensions y and x, but the 25.4 factor appears in the equation because of the difference in measure-
ment units and makes y and x lookas though they are unequal!