Power system fundamentals - Wind Power Integration: Connection and System Operational Aspects - page 29

i

A B

v

Figure 2.4 Simple AC system

The current is taken to lag the voltage by a time f / w , without any loss of
generality. In electrical engineering, this time lag is often expressed as the corre-
sponding angle, f . This reflects the use of phasors, which will be introduced later.
The instantaneous power from A to B is given by the product of voltage and
current as

p ð t Þ¼ V m I m cos w t cos ðw t fÞ

Some rearrangement of the right-hand side of this equation will enable us to
decouple the power into active and reactive components. First, noting that

cos ð A þ B Þ¼ cos A cos B sin A sin B
cos ð A B Þ¼ cos A cos B þ sin A sin B

cos ð A þ B Þþ cos ð A B Þ¼ 2 cos A cos B

∴

and setting A ¼w t and B ¼w t f , we have

p ð t Þ¼ V m I m ð cos ð 2 w t fÞþ cos fÞ= 2

V m I m

p ð cos 2 w t cos f þ sin 2 w t sin f þ cos fÞ
¼ VI ðð 1 þ cos 2 w t Þ cos f þ sin 2 w t sin fÞ
¼ P ð 1 þ cos 2 w t Þþ Q sin 2 w t
Note that V ¼ V m =

p

¼

p :
V and I are effective or root mean square (r.m.s.) values of voltage and current,
respectively. From now on, all values for alternating voltage and current will be
assumed to be r.m.s. unless it is stated otherwise.
The above analysis has introduced two very important quantities:

p and I ¼ I m =

P ¼ VI cos f
Q ¼ VI sin f

ð 2 : 14 Þ

P is the active power , or simply power . This is the average value of the
instantaneous power, which is of course the quantity which power systems are
designed to produce and deliver. The active power depends on f , the angle by
which the current cosinusoid leads or lags the voltage cosinusoid in the analysis.
The term cos f is known as the power factor . It can be thought of as the factor by
which the maximum possible active power for a given voltage and current is
reduced by virtue of current being out of phase with voltage.