The relationship between the power delivered by the generator bus and
voltage at the 2 buses can be represented by the following equations:
PD + jQD =
|V2||I2|ejø
|V2|2 = |V1|2/2
- ßPDX ± [|V1|4/4 - PDX
(PDX + ß|V1|2)]1/2
pf = cos ø
Where:
- PD is the real power and QD the reactive power
delivered to the load bus by the transmission line.
- X is the reactance of the transmission line.
- ø =
Ð
V2 -
Ð I
- ß is tan ø.
The figure below is a plot of |V2| versus PD in
per unit values. This plot assumes a load power factor (pf) of 1.0 (i.e.
ø = 0, ß = tan ø = 0), and |V1| fixed as the reference voltage
with a value of 1 p.u.
Because this power voltage (PV) plot is based on a quadratic equation,
there exists two solutions for |V2| which we will denote as
Pt. A and Pt. B on the plot above. Pt. A represents a stable
operating point for the power system, and Pt. B represents the
reciprocal unstable point where the system cannot and does
not operate.
Voltage Security Assessment seeks to maintain voltage stability
through the off-line analysis of the power system by eliminating any incongruities
before a problem occurs, such as voltage operating below normal bounds.
In this example Pt. A and Pt. B were chosen only to emphasize the nature
of the quadratic equation and its two solutions, at this point we do not
intend to imply anything else with regards to this plot. Later we will
discuss the importance of the position of Pt. A along this PV curve and
the possible system consequences.
It is also cautioned that voltage stability may involve complex dynamics
from generators, voltage controls, and loads. The PV curve concept captures
only some steady-state aspects of voltage stability.
