Power factor correction

The load in Figure 2.7 can be expressed as an admittance Y = G + jB supplied from a voltage V, where Y = 1/Z. G is the conductance, i.e. the real part of the admittance Y, and B is the susceptance, i.e. the reactive or imaginary part of the admittance Y. The load current is I and for an inductive load the reactive component is negative (equation (2.3)) so we can write

${\rm{I}} = {I_R} - j{I_X} = V(G - jB) = VG - jVB$ (2.8)

Both V and I are phasers, and equation (2.8) is represented in the phaser diagram (Figure 2.11) in which V is the reference phaser. The voltage V and current I are in common with Figure 2.9, but Figure 2.11 shows the components of the current I and omits E and the voltage drop across the supply impedance. The load current has a 'resistive' or 'real' component I_R in phase with V, and a 'reactive' or 'imaginary' component, I_X = VB in quadrature with V. The angle between V and I is ϕ, the power-factor angle. The apparent power supplied to the load is given by equation (2 .2) with P = V²G and Q = V²B. For a capacitive load I_X is positive and Q = V²B, which is negative.

The real power P is usefully converted into heat, mechanical work, light, or other forms of energy. The reactive volt-amperes Q cannot be converted into useful forms of energy but is nevertheless an inherent requirement of the load. For example, in AC induction motors it is associated with production of flux and is often called the 'magnetizing reactive power'.

The supply current exceeds the real component by the factor 1/cos ϕ, where cos ϕ is the power factor: that is, the ratio between the real power P and the apparent power S. The power factor is that fraction of the apparent power which can be usefully converted into other forms of energy. The Joule losses in the supply cables are increased by the factor 1/ cos²ϕ. Cable ratings must be increased accordingly, and the losses must be paid for by the consumer.

The principle of power-factor correction is to compensate for the reactive power; that is, to provide it locally by connecting in parallel with the load a compensator having a purely reactive admittance of opposite sign to that of the reactive component of the load admittance. An inductive load is compensated by a capacitive admittance +jB_γ and a capacitive load by an inductive admittance -jB_γ. If the compensating admittance is equal to the reactive part of the load admittance, then for an inductive load the supply current becomes

${{\rm{I}}_{\rm{S}}} = {\rm{I + }}{{\rm{I}}_{\rm{\gamma }}} = V\left( {G - jB} \right) + V\left( {jB} \right) = VG = {I_R}$ (2.9)

Fig. 2.11 Phasor diagram for power factor correction.

which is in phase with V, making the overall power-factor unity. Figure 2.11 shows the phaser diagram.

With 100% compensation the supply current I_s now has the smallest value capable of supplying full power P at the voltage V, and all the reactive power required by the load is supplied locally by the compensator. The reactive power rating of the compensator is related to the rated power P of the load by Q_γ = P tan ϕ. The compensator current Q_γ/V equals the reactive current of the load at rated voltage. Relieved of the reactive requirements of the load, the supply now has excess capacity which is available for supplying other loads. The load may also be partially compensated (i.e. |Q_γ| < |Q|).

A fixed-admittance compensator cannot follow variations in the reactive power requirement of the load. In practice a compensator such as a bank of capacitors can be divided into parallel sections, each switched separately, so that discrete changes in the compensating reactive power may be made, according to the requirements of the load. More sophisticated compensators (e.g. synchronous condensers or static compensators) are capable of continuous variation of their reactive power.

The foregoing analysis has taken no account of the effect of supply voltage variations on the effectiveness of the compensator in maintaining an overall power factor of unity. In general the reactive power of a fixed-reactance compensator will not vary in sympathy with that of the load as the supply voltage varies, and a compensation 'error' will arise. Later the effects of voltage variations are examined, and we will find out what extra features the ideal compensator must have to perform satisfactorily when both the load and the supply system parameters can vary.

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Electric Power Systems