Most Economical Power Factor when kVA demand is constant
Consider an installation having an active power requirement of P_{1} kW at a p.f. of cos φ_{1} . The power triangle at original p.f. cos φ_{1} is OAB as shown in Fig.
Here, OB = P_{1,} OA = S_{1}, angle( AOB) = φ_{1}.
Suppose that power factor is improved to cos φ_{2} by supplying leading kVAr Q_{C} by installing p.f. correction equipment. The kVA output S is to be kept constant. Therefore, the output increases from P_{1} to P_{2}. The power triangle corresponding to this condition is OCD as shown in Fig.
Here, OD = P_{2}, OC = S_{2}, angle(COD) = φ_{2}.
Since the original kVA is equal to the final kVA
OA = OC or S_{1} = S_{2} = S (say)
Increase in active power output of the installation due to improved p.f. = OD — OB = P_{2} — P_{1}
If a = annual cost per kW of installation then the annual saving due to increased power output is given by
C_{P }= a (P_{2} — P_{1}) = a (OD — OB) = a (S_{2} cos φ_{2} —S_{1} cos φ_{1}) = a.S (cos φ_{2} —cos φ_{1})
Reactive kilovoltampere (kVAr) of the p.f. correction equipment is given by
Q_{C} = EC – CD = AB – CD = S1 sin φ_{1 }- S_{2} sin φ_{2 } = S (sin φ_{1} -sin φ_{2})
If b = annual cost per kVAr of the p.f. correction equipment then the annual cost of the power factor correction equipment is given by
C_{PF} = b.Q_{C }= b.S (sin φ_{1} — sin φ_{2})
Net annual saving, C_{S}— C_{P} = C_{PF}
or C_{S }= a.S (cos φ_{2} — cos φ_{1}) — b.S (sin φ_{1} — sin φ_{2}) ……… (i)
For maximum annual saving, C'_{S} = 0 and C''_{S} < 0
It is to be noted that in Eq. (i), a, b, S, φ_{1} are constants and φ_{2} is variable. Differentiating Eq. (i) w.r.t. φ_{2} and equating it to zero we get
—a.S sin φ_{2} + b.S cos φ_{2} = 0
or, tan φ_{2 }= b/a …….. (ii)
= (Annual cost per kVAr of the p.f. equipment)/ Annual cost per kW of installation
The most economical p.f. is cos , where φ_{2} is given by Eq. (ii). In the past, the most economical value of power factor was around 0.95. The present value is towards unity because of the high costs of plant and fuel. However, as the power factor approaches unity, the cost of power correction equipment becomes more. Hence, it is not economical for consumers to raise power factor to unity.