• Most Economical Power Factor when kVA demand is constant

Consider an installation having an active power requirement of P1 kW at a p.f. of cos φ1 . The power triangle at original p.f. cos φ1 is OAB as shown in Fig.

Here,  OB = P1,     OA = S1,    angle( AOB) = φ1.

Suppose that power factor is improved to cos φ2 by supplying leading kVAr QC by installing p.f. correction equipment. The kVA output S is to be kept constant. Therefore, the output increases from P1 to P2. The power triangle corresponding to this condition is OCD as shown in Fig. Here,    OD = P2, OC = S2, angle(COD) = φ2.

Since the original kVA is equal to the final kVA

OA = OC             or S1 = S2 = S (say)

Increase in active power output of the installation due to improved p.f. = OD — OB = P2 — P1

If a = annual cost per kW of installation then the annual saving due to increased power output is given by

CP = a (P2 — P1) = a (OD — OB) = a (S2 cos φ2 —S1 cos φ1) = a.S (cos φ2 —cos φ1)

Reactive kilovoltampere (kVAr) of the p.f. correction equipment is given by

QC = EC – CD = AB – CD = S1 sin φ1 - S2 sin φ2  = S (sin φ1  -sin φ2)

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

CPF = b.QC = b.S (sin φ1 — sin φ2)

Net annual saving, CS— CP = CPF

or       CS = a.S (cos φ2 — cos φ1) — b.S (sin φ1 — sin φ2)             ……… (i)

For maximum annual saving, C'S = 0 and C''S < 0

It is to be noted that in Eq. (i), a, b, S, φ1 are constants and φ2 is variable. Differentiating Eq. (i) w.r.t. φ2 and equating it to zero we get

—a.S sin φ2 + b.S cos φ2 = 0

or,                tan φ2 = 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 φ2 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.