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Unformatted text preview: .SZPW ry "235 “Lonoil Unit 04 Power Factor When a RL load is supplied with a sinusoidal voltage, Figure '1. RL load with supply source ***** u(t)= ‘5 UcosaJt then the supply current can be decomposed into a component
that is inphase with the voltage and a component shifted by
90 degrees i(t) = x5 Icos(a)t— w) =
= ﬁlcosp cosmt + ﬁlsingo sinaJt = = 13.10) + i(t)
The in—phase components is equal to ia(t) =ﬁ1acoswt, Ia=Icos¢=§,
and it is called the active current.
The component shifted by 90 degrees = 1r: Ism¢=EQ]_. is called the reactive current. These two currents are shifted
mutually by 90°, thus the supply current RMS value is equal to
I = JI§+13 = ./(§>2+(—g—)2. When the load has the active power P = U 1 cos qr, then the
supply has to provide the voltage of the RMS value U and the
current of the RMS value I. It means, that the source could be
loadad with the active power U I, but due to the current phase
shift, (0, the active power of the load is equal to P = U 1 cos w.
Therefore, the ratio
_ P
_ W
is referred to as the power factor (PF). It is a measure of the
supply utilization. It can be expressed as [a P
A = =‘an====,
:lPZ + Q2 J]: + Ir2 The presence of the reactive power, Q, with associated
reactive current, 1}, increases the supply current and its RMS
value. Consequently, the power factor, A, declines from unity
value. There are three main harmful effects of low PF, i.e., the
increased supply current RMS value in circuits as shown in
) Figure 1 . These are: 1. Enhanced voltage drop, Au, on the supply source
impedance. 2. = cosw, 2. Enhanced active power loss, AP, on the supply source
resistance. 3. Enhanced required power ratings of the supply equipment
or its reduced capability to supply customers. These three affects of the low power factor make energy
delivery more expensive. More expensive equipment might
be required and more fuel is needed because of the active
power loss in the supply. These three affects of the low power
factor make energy delivery more expensive. More expensive
equipment might be required and more fuel is needed because
of the active power loss in the supply. The difference in the voltage drop in a circuit that at
the same active power, P, and the same load voltage RMS value, U, operates at low PF, 2. = 0.5 or at high PF, 2. = 0.95,
is shown in the diagram of complex RMS (CRMS) values in
Figure 2. Figure 2. Diagrams of CRMS values
(a) at low PF and (b) at high PF ***** Observe that the internal voltage RMS value, E, of the
source that supplies the load with low power factor has to be
higher than the internal voltage of the source that SUpplies the
load with high power factor. To balance the voltage loss, the
intemal voltage RMS value, E, of the supply has to be
enhanced. There are some operational cost related to the load
voltage control by an increase in the distribution voltage. All components of the increased cost of energy delivery at
low power factor are on the side of energy supplier. Thus,
low power factor reduces the supplier’s revenue. Therefore,
power utilities are concerned with the low power factor and
have various policies to enhance it. Power Factor Improvement. Power factor at supply : 'S’upélyséotitée‘ ' Figure 3. Circuit with capacitive compensator
***** terminals can be enhanced by retroﬁtting the load equipment
with equipment that has an enhanced power factor or by
reactive power compensators. Overexited synchronous
motors can serve in threephase systems as such
compensators. In singlephase systems with RL loads only
shunt capacitor banks, connected as shown in Figure. 3, can
by used for such a purpose. The shunt capacitor, C, loads the supply with a reactive
current of the opposite sign than the reactive current of the
RL load, thus it reduces the reactive component of the supply
current. This reduces the phaseshift of the supply current and
its RMS value as shown in Figure 4. Figure 4. Diagram of CRMS values
for compensated load
***** To improve the PF from ,1 = cosq) to zl’= cosqo', the
capacitor should change the reactive current of the supply by
the value IC= mCU=Isingo—Isin(o’. If the load has the active power P = U I A, then the
compensator should have capacitance equal to P U21 [sin(cos'1/1)  sin(cos'1/1,)] .
a) C: In particular, to improve the power factor to unity, the
capacitance should be equal to — P L
wU2/1 wUZ' Beneﬁts of a higher power factor have to be paid by the
investment cost of the compensator. Therefore, a trade off
these beneﬁts and the compensator cost is needed and
consequently, compensators usually have capacitance lower
than C0. Its choice is based on investigation of the active
power and voltage loss in the distribution system versus
compensator capacitance C. The dependence of the active
power loss and the voltage loss in the distribution system on
capacitance C is illustrated in Figure 5. C0 sin(cos'1/1) = Figure 5 show the change of the PF, ,1, power loss and
voltage loss versus capacitance C for a normalized load that
at voltage U = 100V has the active power P = 10 kW at
power factor ,1 = 0.5. The load is supplied from a sinusoidal
voltage source that has the short circuit power 40 times
higher than the load active power P and the reactance to
resistance ratio, Xs/R, = 3. With such parameters and the
frequency normalized to ah = 1 rd/s, the capacitance needed
for total compensation, Co = 1.7 F. Voltage loss in Figure 5 is deﬁned as the difference of
internal voltage, e, and load voltage, u, RMS values, i.e., E 
U. The voltage loss in the supply declines with the
compensating capacitance at an almost constant rate. For
overcompensated loads this voltage loss could even be
negative. One could notice, however, that the voltage loss, E—
U, in the supply source has different meaning than the
voltage drop, EU, on the supply source impedance, since, AU: E—U ¢ E—U=ZSI’. In particular, the voltage drop, Zs I ’, cannot be negative.
our! to “AFR/k3le ‘ 0.9
0.3  Power Factor
0.7
0.5  Voltage Loss
05
DA
03 Q2 Max. Pow. Loss = 316 W 0.1
Max. Volt. Loss = 4.9 V a C u 0.5 1 1.5 2 F
Figure 5. Plot of power factor, voltage loss
and power loss versus compensator capacitance C
$$$$$ D Figure 5 shows that the active power loss can be reduced
more than three times by improving the power factor to unity,
although in the capacitance range around Co the power loss
reduclion is not signiﬁcant. Therefore, to reduce the cost of
the compensator, a capacitance, C, below the value of C0 is
usually selected. ...
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 Fall '08
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