6.7
Relation between PL, QL and V
Neglecting the resistance of generator transformer and transmission line, the equivalent circuit of
the system and its phasor diagram are shown in Fig. 6.11 (a) and (b).
Figure 6.11: Equivalent circuit of the system(a) and the phasor diagram (b)
From the phasor diagram of Fig. 6.11 (b), it can be observed that IX cos φ = E sin δ and
IX sin φ = E cos δ − V
IX cos φ EV
=
sin δ
X
X
PL (V ) = V I cos φ = V
(6.177)
QL (V ) = V I sin φ = V
2
IX sin φ EV
V
=
cos δ −
X
X
X
The angle δ between Ē and V̄ phasor can be eliminated using the identity sin2 δ + cos2 δ = 1
and thus one can obtain:
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EV 2
V2 2
2
[
] = [PL (V )] + [QL (V ) +
]
X
X
(6.178)
This static power-voltage equation determines all the possible network solutions when the voltage
characteristics PL (V ) and QL (V ) are taken into account.
For an ideally stiff load, the power demand of the load is independent of voltage and is constant
PL (V ) = PL and QL (V ) = QL , where, PL and QL are the real and reactive power demand of the
load at the rated voltage V.
For stiff load the equation (6.178) can now be written as:
EV 2
V2 2
2
[
] = [PL ] + [QL +
]
X
X
(6.179)
Substituting QL = PL tan φ in equation (6.179) one can obtain,
EV 2
V2 2
V2
=[
] −[ ]
P + P tan φ + 2 PL tan φ
X
X
X
2
L
Substituting tan φ =
2
L
2
(6.180)
sin φ
and noting that sin2 φ + cos2 φ = 1 equation (6.180) can be further
cos φ
simplified as
V2
V2 2
P + 2 PL
sin φ cosφ = 2 (E − V 2 ) cos2 φ
X
X
2
L
(6.181)
2
V2
Adding and subtracting (
sin φ cos φ) to the right hand side of equation (6.181), gives
X
2
V2
V2 2
V2
(PL +
sin φ cos φ) − ( ) sin2 φ cos2 φ = ( 2 ) (E 2 − V 2 ) cos2 φ
X
X
X
After simplifications the equation can be expressed as:
PL +
√
V
V2
sin φ cos φ =
cos φ E 2 − V 2 cos2 φ
X
X
(6.182)
The voltage at the load bus can be expressed in per unit as V/E. Equation (6.182) can be
expressed as:
E2 V 2
E2 V
PL = − ( ) ( ) sin φ cos φ + ( ) ( ) cos φ
X
E
X
E
Or
Where p =
√
V 2
1 − ( ) cos2 φ
E
√
p = −v 2 sin φ cos φ + v cos φ 1 − v 2 cos2 φ
(6.183)
PL
V
, and v =
2
(E /X)
E
Equation (6.183) describes a family of curves with φ as a parameter. One such P-V curve is
shown, for a particular value of power factor cos φ in Fig. 6.12.
The Power-Voltage curve (PV-curve) presents load voltage as a function of load real power. For
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Figure 6.12: PV curve for lagging power factor load
static load (PL =constant) as shown in the figure, two operating points (A) and (B) are possible.
Point (A) represents low current high voltage solution and is the desirable operating point, while
point (B) represents high current low voltage solution. Operation at point B is possible, although,
perhaps non-viable due to low voltage and high current condition. Further, with system initially
at ‘point A’, if the load is increased then from the curve it can be seen that the voltage will drop.
Increase in load will result in an increase in the current flowing in the transmission line, hence the
voltage will reduce and this is a perfectly normal response of the system. Hence, point A and the
upper portion of PV curve represent stable system operation region. At point B, however, as the
load is increased the system voltage increases which is not possible at all. Hence, point B and the
lower portion of PV curve represent unstable operating region. Power systems are operated in the
upper part of the PV-curve. As the load increases point (A) and (B) come closer and coincide at
the tip of P-V curve. This point is called the maximum loading point or critical point. Further
increase in the load demand results in no intersection between the load-characteristic and PV curve
and hence, represents voltage instability this is shown in Fig. 6.13.
The impact of large a disturbance on voltage stability can also be explained with the help of PV
curves. Suppose, a large disturbance causes the loss of a transmission line resulting in increases in
reactance X or loss of generator resulting in reduction in E. The post-disturbance and pre-disturbance
PV characteristics along with load characteristic are shown in Fig. 6.14. The large disturbance causes
the network characteristic to shrink drastically, so that the post disturbance PV curve and the load
characteristic do not intersect at all. This causes voltage instability leading to a voltage collapse.
Assuming a smooth increase in load, the point where the load characteristic becomes tangent to
the network PV characteristic defines the loadability limit of the system. Any increase in load beyond
the loadability limit results in loss of voltage stability, and system can no longer function. In Fig.
6.13, the point where the load characteristic is tangent to network PV curve, coincides with the maximum deliverable power for a constant power load. However, a loadability limit need not necessarily
coincide with the maximum deliverable power, as it is dependent on the load characteristic.
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Figure 6.13: Changes in the operating point wih increasing load
Figure 6.14: Loss of voltage stability due to a large disturbance
Figure 6.15: Maximum deliverable power and loadabilty limit for polynomial load
V α
This is shown in Fig. 6.15 for a polynomial load P = P0 ( ) , where, P0 represents the base
V0
value of load active power at rated voltage V0 = 1 p.u. and α represents the voltage exponent. α = 0
represents a constant power load.
Equation (6.183) when plotted for different values of φ gives a family of PV curves. Because of
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Figure 6.16: PV curve drawn for different values of power factor
their characteristic shape, these curves are referred to as nose curves. The following observations
can be made regarding the curves shown in Fig. 6.16:
• For a given load below the maximum, there are two possible solutions- one with higer voltage
and lower current and the other with lower voltage and higher current. The former corresponds
to ’normal’ operating conditions, with the load voltage V closer to the generator voltage.
• As the load is more and more compensated (corresponding to smaller tan φ ), the maximum
deliverable power increases, and the voltage at which this maximum occurs also increases.
• For over-compensated loads (tan φ < 0 , leading power factor), there is a part of the upper
PV curve along which the voltage increases, as the load power increases. This can be explained
as follows: when tan φ is negative then, with more active power consumption more reactive
power is produced by the load. At low values of load, the voltage drop due to increased active
power is offset by the increase in voltage due to increased reactive power. The more negative
tan φ is, the larger is the portion of PV curve where this voltage rise occurs.
The usefulness of the nose curve is high in practice as the difference between a particular load
and maximum load, determined by the peak of the characteristic, is equal to stability margin for a
given power factor.
√
From equation (6.183), when QL = 0 ( and hence φ = 0), p = v 1 − v 2 . To find the value of PL
dp
= 0 and the solution of the resulting equation gives the values
dv 2
1
1
E
E2
of v = ± and p = . Hence, PL =
. Note that
is the short circuit power at the load bus,
2
2
2X
X
at the peak of the nose curve set
as it is the product of no load voltage E and the short circuit current (E/X). The maximum power
limit for a lossless line, with unity power power, thus corresponds to half the short circuit power.
PV-curves (nose curves), illustrate the dependency of the voltage on real power of a composite
load assuming that the power factor is a parameter. The QV curves discussed next are derived
assuming that the voltage is a parameter.
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For given value of V, equation (6.179) describes a circle in the (PL − QL ) plane as shown in Fig.
6.17 (a). The centre of the circle lies on the QL -axis and is shifted vertically down from the origin
by (V 2 /X) , the radius of the curve is (EV /X) . Increasing the voltage V produces a family of
circles of increasing radius and increasing downward shift, bounded by an envelope as shown in Fig.
6.17 (b).
Figure 6.17: QP curves for stiff load (a) one circle for a given V (b) family of curves for different
voltages and their envelope
For each point inside envelope, for example, point A, there are two possible solutions to equation
(6.179), at voltages V1 and V2 , as it lies on both the circles. For any point on the envelope, say point
B, there is only one value of V for which the equation (6.179) is satisfied. By determining the values
of PL and VL for which only one solution to equation (6.179) exists, the equation of the envelope
can be developed. Rearranging equation (6.179), one gets:
V2 2
E2 V 2
( ) + (2QL −
) ( ) + (PL2 + Q2L ) = 0
X
X
X
(6.184)
This quadratic equation in (V 2 /X) has only one solution when
E2 2
D = (2QL −
) − 4 (PL2 + Q2L ) = 0
X
(6.185)
Solving for QL one gets
QL = (
P2
E2
)−[ 2L ]
4X
(E /X)
(6.186)
This represents the equation of an inverted parabola that crosses PL axis at E 2 /2X and has its
maximum at
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PLmax = 0; QLmax =
E2
4X
(6.187)
Thus the maximum reactive power supplied to a load with PL = 0 i.e. a purely reactive load
is equal to one fourth of the short-circuit power. Also a point with coordinates PL = E 2 /2X and
QL = 0 corresponds to the peak of the nose curve for φ = 0 and QL = 0.
The parabola as described equation (6.186) defines the shape of envelope in Fig. 6.17 (b) that
encloses all the possible solutions to the network equation (6.179). Every point (PL , QL ) inside the
parabola satisfies two possible network solutions corresponding to two distinct values of load voltage
V, and each point on the parabola satisfies only one network solution corresponding to only one
possible value of voltage. Further, there are no network solutions outside the parabola implying that
it is not possible to deliver power for a (PL , QL ) point lying outside the parabola.
Various stability criteria to assess the system voltage stability are discussed in the next lecture.
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