The definitions of reactive power dealt with in Sects. 7.1.7 and 11.2 can be augmented by a general statement. So long as the voltage and current waveforms are in phase the reactive power is zero. Phase displacements between voltage and current act as reactive power even when pure resistors are supplied via the converter.
This applies generally even with non-sinusoidal ac voltage [11.1]. Reactive power always occurs with converters since the instantaneous values of voltage and current normally are not linearly related. However, this reactive power can, if necessary, be kept low by appropriate methods (e.g. by the sector control dealt with in Sect. 8.3.5). It is desirable that the power factor, determined by the ratio of transmitted active power to apparent power arising should be as high as possible.
The optimum limiting value attainable for the power factor A is therefore 1 [11.9, 11.10].
11.7.1 Reactive Power Compensation
The reactive power in single-phase and three-phase ac circuits can be compensated by supplementary reactances (usually capacitive). Such reactances can be contactlessly switched by semiconductor switches (see Sect. 6.1.4). Using semiconductor switches inductive rectances can also be continuously adjusted by phase control (see Sect. 6.2). In addition converters can be employed as pure reactive power converters whereby their inductive or capacitive reactive power consumption can be continuously varied [11.6, 11.12].
Besides the undesired loading of systems and installations by reactive power reactive currents generate voltage drops across the system reactances. The voltage drops caused by uneven loads such as welding machines are furnaces or drives with surge-like current consumption lead to system voltage fluctuations called
"flicker". When the pulsations of reactive power occur in the range of several Hz, the light fluctuations caused by flicker are tiring to the human eye.
Reactive Power Compensation and Balancing of Unbalanced Load 231
IV
jUN uvj IK ~ . T~
Line
'--v----'
Compensation equipment
Without Reactive current Voltage compensation control control
Fig. 11.20. Circuit diagram and phasor diagram of a resistive-inductive load with capacitive power factor compensation
Reactive power fluctuations can be compensated by converters or by reactances switched by power semiconductors. Not only inductive but also capacitive reactances can be switched by power semiconductors connected in antiparallel. The maximum switching frequency causing no transient oscillations equals twice the system frequency. This result in good dynamics [11.26, 11.27, 11.29].
Figure 11.20 shows the principle of parallel compensation of the voltage drop in a single-phase ac system with inductive and resistive internal impedance by a thyristor-switched compensating capacitor. With pure power factor compens- ation the resistive voltage drop remains. Compensation of the system current to the system current line inclined at an angle of <PN = arc tan (XN/RN) also reduces the resistive voltage drop.
If the compensating capacitors are binary stepped, four capacitor stages can produce 15 different steps of reactive power compensation (Fig. 11.21). In this case capacitors are connected via thyristors with diodes in antiparallel. When semicontrolled switches are employed, the maximum switching frequency drops to the system frequency. Nevertheless, good dynamic voltage stability can be attained by this method.
Instead of with compensating capacitors it is also possible to work with switched compensating inductances. In this case stepped reactors are switched in
a b
Fig. 11.21a,b. Dynamic voltage stabilization with thyristor- switched power capacitors. a Plant;
b power factor compensation equipment
and out via thyristors in antiparallel. Since the switching in of inductances increases the inductive reactive current, supplementary fixed compensating capacitors must be provided.
11.7.2 Balancing of Unbalanced Load
When multi-phase ac systems are loaded unbalanced load can occur i.e. different currents in the individual phases. In this case the individual phases are unsymmetrically loaded and power pulsations occur even on otherwise balanced multi-phase systems.
According to Steinmetz a single-phase resistive load between two conductors of a three-phase system can be balanced by means of capacitive and inductive reactances. Figure 11.22 shows the conditions with load balancing in a three-phase ac system.
The capacitive and inductive reactances required for balancing (Fig. 11.22a) can be calculated in accordance with equation
(11.47 ) With these reactances symmetrical resistive currents II, 12 , and 13 arise in the three-phase ac system.
When in addition inductive loads are to be compensated (Fig. 11.22b) the required compensating capacitors can be calculated in accordance with the
a
T 2 3
b
...---.
.-- - r:;;
"
II
Load
2 3
ion Compensat
c
...---.
- -
R'2 -
...r--1-
..r-""1.
- T;j -
Jl II
II II C~3
"
Fig. 11.22a-c. Complete reactive power compensation on unsymmetrical load. a Load balancing in accordance with Steinmetz; b reactive power compensation; c reactive power compensation with load balancing
Reactive Power Compensation and Balancing of Unbalanced Load 233
equation
(11.48 ) If reactive power compensation with load balancing is to be carried out for all three phases (Fig. 11.22c), the capacitors required can be calculated in accordance with the equation
(11.49 ) The capacitors for the other phases are obtained by rotating the indices. Under certain conditions Eq. (11.49) can produce a negative value. In this case instead of a capacitor an additional inductance is required.
Figure 11.22c shows how by combination of compensating capacitors between the three conductors of a three-phase ac system an unbalanced resistive-inductive load can be balanced and completely power factor corrected.
The coupling of ac systems which have different number of phases represents a similar problem. Figure 11.23 shows how the unbalanced loading of the three- phase ac system illustrated in part a can be balanced by appropriately arranged inductive and capacitive reactances whereby an inductive component of the current 112 in the single-phase system can be eliminated by a supplementary capacitor CK • When the current in the single-phase system is variable the
I; 12 13
*
12=-1'2
~'2 U,2 • 13=0 !' I
I, =1,2 1,2
1,2 U2 U,
a b
Fig. 1l.23a,b. Reactive power compensation and balancing when out of phase ac systems are coupled at constant frequency. a Unsymmetrical loading; b reactive power compensation and balancing
reactances must be adjusted accordingly. This can be done with the aid of semiconductor switches. The reactances can in such a case again be binary graded and switched in and out via thyristor switches in synchronism with the supply without transient phenomena.
Instead of with reactances the balancing and power factor correction of a three-phase system can also be carried out using a reactive power converter.