Relationships between power, reactive power, voltage levels and load angle

The phasor diagram for the system in Figure 2.16 is shown in Figure 2.17, assuming that the load has a lagging power factor angle ϕ. The line or cable is represented by its impedance

, and

is again neglected (being usually much smaller than

. The voltage drop across the transmission line is j

, which leads the phasor I by

. The angle between

and

is the load angle, δ and

and

(2.17)
Also

(2.18)
From this we get the power flow equation

(2.19)

and the reactive power equation for the receiving end

(2.20)
Evidently

as long as the transmission losses are negligible. At the sending end,

(2.21)

Fig. 2.17 Phasor diagram for Figure 2.16.

from which it can be shown that

(2.22)

Note the symmetry between this expression and the one for

in equation (2.20).
Example 1

Suppose

p.u. and

p.u.(The per‐unit system is explained in Section 2.13. If you aren’t familiar with it, try to read these examples as practice in the use of normalized (per‐unit) values. In effect, they make it possible to forget about the units of volts, amps, etc.) The transmission system has

. and

is negligible. We have

and

. Then

p.u. and

p.u. Thus the receiving end is generating reactive power and so is the sending end. The power factor is lagging at the sending end and leading at the receiving end. The phasor diagram is shown in Figure 2.

(not to scale).

Example 2

Suppose

p.u. while the receiving‐end voltage is reduced to

p.u., with

. and

. Now

and δ

, slightly larger than in example 1 because

is reduced by 5%. Also

. and

p.u. Since

is positive, the sending‐end generator is generating VArs.

is also positive, meaning that the receiving‐end load is absorbing VArs. The phasor diagram is similar to that shown in Figure

(but not to the same scale).

Notice that a 5% reduction in voltage at one end of the line causes a massive change in the reactive power flow. Conversely, a change in the power factor at either end tends to cause a change in the voltage. A 5% voltage swing is, of course, a very large one. Changes in the power tend to produce much smaller changes in voltage; instead, the load angle δ changes almost in proportion to the power as long as δ is fairly small