101
CHAPTER 5
INSTANTANEOUS REACTIVE POWER THEORY
5.1
INTRODUCTION
This chapter presents a novel control strategy for the case of three
phase four wire Unified Power-Quality Conditioner (UPQC) based on the
concepts of instantaneous active and reactive Power theory. Using
instantaneous a c t i v e a n d reactive Power theory, current harmonics,
reactive
power
compensation and voltage c o m p e n s a t i o n have been
simulated and the results are analyzed. The operation and capability of the
proposed system was analyzed through simulations with MATLAB /
SIMULINK.
5.2
REACTIVE POWER THEORY
The Generalized Theory of the Instantaneous Reactive Power in
Three-Phase Circuits also known as instantaneous power theory , or p-q
theory. Instantaneous Reactive Power Theory is based on set of instantaneous
powers defined in the time domain and uses the Park Transform. No
restrictions are imposed on the voltage or current waveforms and it can be
applied to three-phase systems with or without a neutral wire for three-phase
generic voltage and current waveforms. Thus, it is valid not only in the steady
state but also in the transient state. This theory is very efficient and flexible in
designing controllers for power conditioners based on power electronics
devices (Akagi et al 2004).
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The method offers the technique to calculate the real and reactive
power requirements of the load instantaneously. The method is mostly applied
to calculate the reference current of the shunt and series active filter. The p-q
theory consists of an algebraic transformation (Clarke transformation) of the
three-phase voltages and currents in the a-b-c coordinates to the
-0
coordinates, followed by the calculation of the p-q theory instantaneous
power components.
The p-q theory is one of several methods that can be used in the
control active filters. It presents some interesting features, namely,
It is inherently a three-phase system theory
It is based in instantaneous values, allowing excellent dynamic
response
It can be applied to any three-phase system (balanced or
unbalanced, with or without harmonics in both voltages and
currents)
Its calculations are relatively simple (it only includes algebraic
expressions that can be implemented using standard
processors)
It allows two control strategies: constant instantaneous supply
power and sinusoidal supply current
In three-phase circuits, instantaneous currents and voltages are
converted to instantaneous space vectors. In instantaneous power theory, the
instantaneous three phase currents and voltages are calculated as given in
following equations. These space vectors are easily converted into the
orthogonal coordinates (Akagi et al 2004).
103
1
1
1
2
V0
2
3
V
V
2
1
1
0
3
1
1
2
3
i
i
2
2
Va
2
Vb
3
2
(5.1)
Vc
2
1
2
i0
1
2
1
1
0
3
2
2
2
1
2
3
2
ia
(5.2)
ib
ic
The instantaneous active and reactive power in the
-0
coordinates are calculated with the following expressions:
P0
where
V0 I 0
p V I
V I
q V I
V I
(5.3)
p0 = instantaneous zero-sequence power
p = instantaneous real power
q = instantaneous imaginary power
The values of p and q can be expressed in terms of the dc
components plus the ac components, that is,
p
p
p
(5.4)
q
q q
(5.5)
The power components p and q are related to the same
and currents, and can be written together as given below:
voltages
104
p
V
V
i
q
V
V
i
V I
and V I are instantaneous real
(5.6)
and imaginary powers
respectively. Since these equations are products of instantaneous currents and
voltages in the same axis, in three-phase circuits, instantaneous real power is
p and its unit is watt. In contrast V I and V I are not instantaneous powers.
Since these are products of instantaneous currents and voltages in two
orthogonal axes, q is not conventional electric unit like watt or VAr. The
value q is instantaneous imaginary power and its unit is imaginer volt ampere.
These power quantities given above for an electrical system represented in ab-c coordinates and have the following physical meaning
p
-
mean value of the instantaneous real power – It is corresponds to
the energy per time unity which is transferred from the power
supply to the load, through the a-b-c coordinates, in a balanced way
(it is the desired power component). It is the dc component of the
instantaneous power p, and is related to the conventional
fundamental active current.
p
-
alternated value of the instantaneous real power – It is the energy
per time unity that is exchanged between the power supply and the
load, through the a-b-c coordinates. It is the ac component of the
instantaneous power p, it does not have average value, and is
related to the harmonic currents caused by the ac component of the
instantaneous real power.
q
-
instantaneous imaginary power – It is corresponds to the power that
is exchanged between the phases of the load. This component does
not imply any transference or exchange of energy between the
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power supply and the load, but is responsible for the existence of
undesirable currents, which circulate between the system phases
q
-
dc component of the imaginary instantaneous power q, and is
related to the reactive power generated by the fundamental
components of voltages and currents.
q
-
ac component of the instantaneous imaginary power q, and is
related to the harmonic currents caused by the ac component of
instantaneous reactive power. This component does not imply any
exchange of energy between the power supply and the load, but is
responsible for the existence of undesirable currents, which
circulate between the system phases
The objective of the p-q theory is to make the source to deliver the
constant active power demanded by the load. At the same time the source
should not deliver any zero sequence active power. The reference source
current in the
-0 frame is therefore:
ic*
1
ic*
v2
v
v
v2 v
v
.
(5.7)
Since the zero-sequence current must be compensated, the
reference compensation current in the 0 coordinate is i0 itself
ico*
i0
(5.8)
In order to obtain the reference compensation currents in the a-b-c
coordinates the inverse of the transformation given in Equation (5.2) is
applied.
106
1
2
*
ica
i*cb
*
icc
2
3
1
1
1
2
1
0
2
3
ic*0
2
ic*
(5.9)
*
1
2
2
3
ic
2
The compensation current for neutral is given by:
*
icn
*
ica
icb*
icc*
(5.10)
The calculations presented so far are synthesized in Figure 5.1 and
correspond to a shunt active filter control strategy for constant instantaneous
supply power.
Figure. 5.1 Calculation for the constant instantaneous supply power
107
5.3
CIRCUIT CONFIGURATION OF UPQC
The UPQC shown in Figure 5.2 consists of two VSCs (VSC_1 and
VSC_2) that are connected back to back through a common energy storage dc
capacitor . Series converter (VSC_1) is connected through transformers
between the supply and point of common coupling (PCC). Shunt converter
(VSC_2) is connected in parallel with PCC through the transformers. VSC_1
operates as a voltage source while VSC_2 operates as a current source . The
power circuit of VSC_1 consists of three single-phase H-bridge voltagesource PWM inverters. H-bridge inverters are controlled independently.
The main objective of VSC_1 is to mitigate voltage disturbances
originating from supply side. The ac filter inductor and capacitor are connected
in each phase to prevent the flow of harmonic currents generated due to
switching. The transformers connected at the output of each H-bridge inverter
provide isolation, modify voltage/current levels, and prevent the dc capacitor
from being shorted due to the operation of various switches. The power circuit
of VSC_2 consisting of a three-phase voltage-source PWM inverter is supplied
from dc link . VSC_2 is directly connected through a boost inductor Lsm
which can boost up the common dc link voltage to the desired value.
The objectives of VSC_2 are to regulate the dc link voltage
between both converters and to suppress the load current harmonics. The
switching devices in VSC_1 and VSC_2 are insulated-gate bipolar transistors
(IGBTs) with antiparallel diodes. A three-phase uncontrolled diode-bridge
rectifier with resistive and inductive load is used to produce harmonic current.
The ac reactor Lch placed before the rectifier to enhance the load impedance.
Operation modes of UPQC as follows.
108
Figure 5.2 Schematic diagram of UPQC
1)
VSC_1 off and VSC_2 on: When the PCC voltage is within
its operation limits, VSC_1 is closed and VSC_2 works as the
current source . During this operation of UPQC, two lower
IGBTs of each phase H-bridge inverter of VSC_1 remain
turned on while the two upper IGBTs remain turned off,
forming a short circuit across the secondary (inverter side)
windings of the series transformer through Lf.VSC_2
suppresses the load current harmonics and regulates dc-link
voltage during this mode of operation.
2)
VSC_1 on and VSC_2 on: When the PCC voltage is outside
its operating range, both VSC_1 and VSC_2 are open.VSC_1
starts to mitigate disturbances using the energy stored in VDC
and VSC_2 continue to suppress the load current harmonics
and to regulate dc-link voltage.
109
5.4
UPQC CONTROL STRATEGY
A UPQC is the extension of the unified power-flow controller
(UPFC) concept at the distribution system. It consists of combined series and
shunt converters for simultaneous compensation of voltage and current
imperfections in a point of common connection (PCC) in the distribution side.
The series and shunt converters connected back-to-back via a common DC
link capacitor. Unlike the UPFC, the series converter is connected to a supply
side and shunt converter is connected load side. This configuration has proved
performance with both supply voltage distortions such as harmonics and
unbalanced line to line voltages as well as load disturbances such as harmonic
current, unbalanced load and reactive power requirement by the load. This
configuration also provides optimum rating for a specific amount of reactive
power compensation.
UPQC is controlled in such a way that the voltage at the load bus is
always sinusoidal with the desired magnitude. Therefore the voltage injected
by series APF must be equal to the difference between the PCC voltage and
the desired load voltage. The series APF acts as a controlled voltage source.
The shunt APF acts as a control source for maintaining the DC link voltage.
The shunt APF also provides required var to the load such that the power
factor at PCC is unity and only fundamental active power is supplied by the
source. The voltage injected by series APF can be varied from 0 to 360
degree.
The control system of the UPQC can be divided into two parts,
namely a shunt APF controller and a series APF controller, in which they
control the shunt current and the series injected voltage respectively. Source
side inverter, called the series inverter is connected through coupling
transformers between the point of common connection and load. The load
110
side inverter, called the shunt inverter is connected in parallel through the
transformers or directly connected.
The series inverter operates as a controlled voltage source, while
the shunt inverter operates as a controlled current source. The method
proposed in the chapter can be used to simultaneously calculate the harmonics
present in current and voltage waveforms, aiming the application in the
control of a UPQC.Therefore, measurement and extraction waveform
compensation components of UPQC are of great significance for its normally
performance. The series injected voltage has to be in phase with PCC voltage
to compensate voltage sag. The series connected inverter injects a voltage in
quadrature
with the line current thereby emulating an inductive or a
capacitive reactance in series with the line.
The shunt connected inverter injects a reactive current, thereby also
emulating a reactance at the point of connection. While operating both series
and shunt connected inverters together as UPQC, the reactive power supplied
from source reduces, thus increasing the power factor at PCC. The load
voltage is also compensated against power quality disturbances. The UPQC
controller was designed using the instantaneous power method based on
0
transform and fundamental positive sequence detection.
The UPQC under study in Figure 5.2 is analyzed with p-q-r
instantaneous reactive power theory. In the p-q-r theory proposed,p-q-r
reference frames which rotates according to the voltage space vector of a
three-phase four-wire system .In this theory, the current and voltage in space
coordinate are transformed to p-q-r coordinates. Voltage components only in
p axis, which can simplify the active and reactive current calculation. The
current space vector is transformed into three linearly independent dc based
components, in which p-axis component represents the instantaneous active
111
power when multiplied with the single voltage vector component, where the
q-axis component represents an imaginary current component on
plane,
and r-axis component represents an imaginary component which is highly
related to the 0-axis in
-0 reference frames.
A control strategy with the combination of voltage detection
method for series active filter and current detection method for shunt active
filter is used, with the reference signal determined by the voltage detection
and current detection method in p-q-r theory. Since the current is
compensated to be a constant length vector that is rotating aligned with the
voltage space vector, when the load voltage cannot fully be compensated to
be balanced and sinusoidal, the current rotating with the voltage space vector
will not be sinusoidal too.
An extra q-axis component is proposed to add to the reference
source current space vector to force it to rotate with a pure sinusoidal
reference. The power supplied by the series active filter and loss in switching
devices is obtained by drawing extra active power from the parallel active
filter. With the aid of p-q-r theory, a control block model of the integration
feedback is formed and the amount of power is determined by an integration
feedback of dc storage power. The equivalent model of integration controller
is proposed by detecting the instantaneous value of dc storage capacitor
voltage. The simulation with MATLAB verifies that the proposed analysis
and control algorithm is valid for the UPQC under three-phase four-wire
system.
The control system of UPQC has three major elements, which are a
positive sequence detector, a shunt inverter control, and a series inverter
control. The positive-sequence detector extracts the positive sequence of
component from the disturbed and unbalanced three-phase source voltage
112
with series of steps as given in the Figure 5.3.The transformed positive
sequence reference voltage V , V , based on the
0 transform are found
out as explained below.
Figure 5.3 Shunt inverter control
The measured source voltage passes through the three phase Phase
Locked Loop(PLL) and the sine wave generator to calculate the fundamental
component of the
transformed current are i =sin 1t and i =cos 1t.The
powers corresponds to positive sequence fundamental component are
calculated as active power ps and reactive power qs from the of the source
voltage Vs and fundamental current components i and i .
V
V
1
2 1
3 0
p s'
V
q s'
V
3
V
V
1
2
2
2
3
Vsa
Vsb
2
(5.11)
Vsc
i'
i'
(5.12)
113
So, the instantaneous value of the positive-sequence component
voltage is,
v s'
1
v s'
i '2 i ' 2
i'
i'
i'
i'
i'
p s'
i'
qs'
(5.13)
The instantaneous active power loops in the UPQC system and
imaginary power provided by the source should be reduced to a smaller value
such that a larger capacity of power is available to the load side.
5.4.1
Shunt Inverter Control
When there are harmonics and negative sequence components in
the load voltage, the p-q-r reference frames are not rotating at a constant
angular velocity and thus the resulting source current is not sinusoidal. Here, a
compensation method is proposed to control the reference current space
vector to follow the rotation of the sinusoidal reference wave. Since the p-axis
and r-axis are used to compensate the active power harmonics and neutral line
currents respectively, and the remaining y-axis is always located on the
plane, it can be used to move the current space vector to be aligned with the
reference wave, without using extra active power.
For the reference current determination, the load voltage is taken as
the p-q-r rotation reference instead of the reference load voltage since this
guarantees the shunt active filter to compensate the load current harmonics
and to provide the correct amount of power looping in the UPQC, even if the
series active filter cannot fully compensate for the harmonics and unbalance
occurs in the source voltage. Since the p-q-r theory does not require voltage
preprocessing, the control algorithm still works under this case. Unlike a
conventional shunt active filter, it should absorb active power to provide the
114
proper operation of shunt active filter, such that the average of the dc
capacitor voltage is zero.
When the voltage is not balanced and sinusoidal, the instantaneous
frequency of rotation of the p-q-r reference frames is not constant, thus there
is a deviation between the current reference and the ideal voltage reference.
The q-axis component is used to force the original current vector to be
aligned with the balanced and sinusoidal reference wave. The dc component
which corresponds to the average active power component of the load current,
is extracted, and a small amount of r-axis current is added to compensate the
neutral current.
The functions of the shunt inverter are to compensate the current
harmonics, the reactive power, and to regulate the DC link capacitor voltage.
Figure 5.3 shows the configuration of shunt inverter control, which includes
the current control for harmonic compensation and the DC voltage control.
Shunt control calculates the reference value of the compensating current for
the harmonic current and the load reactive power.The instantaneous power is
calculated using
components of positive sequence voltage and load
current iL.
p
vs'
q
vs'
ic*
'
1 vs
U vs'
ic*
where
U
vs' 2
vs'
iL
vs'
(5.14)
iL
vs'
vs'
P ploss
(5.15)
q
v s' 2
Power corresponds to harmonic content is calculated by separating
oscillating power and fundamental power by passing 5th order butter-worth
high pass filter. Using these active powers (oscillating power and system
115
power loss) and reactive power the reference value of the compensating
current is derived from Equation 5.15.
5.4.2
Series Inverter Control
Figure 5.4 shows the control circuit of series converter. The
function of the series inverter is to compensate the voltage disturbance such as
voltage harmonics, voltage unbalance on the source side, which is due to the
fault and/or line drop because of over load in the distribution line.
Figure 5.4 Control circuit for Series converter
The series active filter power loss, includes the switching loss in the
inverters and the copper loss in the coupling transformers. When there is
voltage flicker, or the voltage is below the rated voltage, ac and dc active
power is needed to draw from the series inverter to the load. The series
inverter control calculates the reference voltage to be injected by the series
inverter, comparing the positive-sequence component Vabc’ with the
disturbed source voltage Vsabc.
The v o l t a g e compensation may involves supplying / absorbing
real power from the supply line, so there must be real power balance
between series and shunt
inverters . The instantaneous real power
absorbed / delivered by the series inverter must be equal to the real power
delivered / absorbed by the shunt inverter so as to maintain DC link
capacitor voltage constant. In the Figure 5.4, k=(Vrms/sqrt(3))*sqrt(2) and
G=desired maximum phase voltage value. The reference load voltage of a
116
UPQC system is taken to be balanced and sinusoidal as for the requirement of
sensitive loads, however, the fundamental phase angle between the source
voltage and load voltage can be varied and so as the phase angle of source
current with respect to the load voltage. Varying the phase angles results
approximately same amount of power supplied by the source, however, the
magnitude of the source current and the magnitude of series inverter voltage
varies.
The dc component in r-axis imaginary power refers to the reactive
power component in the load, and the ac component is related to the
harmonics and imbalance. The q-axis imaginary power is mainly produced by
the neutral current imbalance. Reducing the amount of imaginary power
supplied by the source results an increase of capacity of active power
available to the load side. The imaginary power can be fully compensated by
the parallel active filter while the series active filter supplies an imaginary
power with a zero average vector sum to compensate the imbalance and
harmonics in the source voltage.
In order to provide a maximum available capacity for the load side,
the power drawn by the series active filter from the dc link can be reduced by
selecting the reference load voltage to be in phase with the fundamental
positive sequence of source voltage, and the current reference is selected to
eliminate the load current's harmonics, imbalance and reactive power, and
includes an active power for the power given out by the series active filter.
To obtain the voltage and current references as analyzed above,
fundamental positive sequence of source voltage is needed to be extracted,
and the load current components corresponding to the harmonics, imbalance
and reactive power must be extracted. By this algorithm, a balanced and
sinusoidal reference wave can be obtained without time delay, and can be
maintained even if the source voltage is distorted by severe voltage sags.
117
IGBTs are used in the inverter circuits to convert DC power to AC
power. A voltage source inverter is energized by a constant DC voltage
supply of low impedance at the input. The output voltage is independent of
load current. The inverters are then connected in series to the distribution line
through injection transformers. IGBT is a unidirectional conducting device
and hence in most of the applications an anti-parallel diode has to be used.
When IGBTs are used as switching components in an inverter or converter,
freewheeling diodes are needed to sustain the current from the inductive load
such as a motor or transformer.
PWM switched inverters provide superior performance to control
asymmetries and especially over currents during unbalanced faults. Use of
single-phase H-bridge PWM inverters in DVR power circuit makes possible
the injection of positive, negative and zero sequence voltages. The voltage
control is achieved by modulating the output voltage waveform within the
inverter. The main advantage of PWM inverter is including fast switching
speed of the power switches. PWM technique offers simplicity and good
response. Besides, high switching frequencies can be used to improve on the
efficiency of the converter, without incurring significant switching losses.
LC filter suppresses the dominant harmonics produced by inverter
circuit. In this thesis Inverter side filtering is preferred for harmonic
elimination. This scheme has the advantageous of being closer to the
harmonic source and low voltage side thus it prevents the current harmonics
to penetrate into the series injection transformers.
The transformers reduce the voltage requirement of the inverters
and provide isolation between the inverters. This can prevent DC storage
capacitor from being shorted through switches in different inverters
(Ghosh et al 2002). The electrical parameters of series injection transformer
should be selected correctly to ensure the maximum reliability and
118
effectiveness. In normal bypass mode, full load currents pass through
semiconductor switches.
5.5
SIMULATION RESULTS
In this section, MATLAB/SIMULINK software simulation results
are presented to show the performance of UPQC and analyzing the operation
of the UPQC for mitigation of current and voltage harmonic elimination . The
power circuit is modeled as a 3- phase 4-wire system with a non-linear load
that is composed of a 3-phase diode-bridge rectifier with RL load. UPQC uses
common DC link for APF and DVR . The performance of VSC_1 and VSC_2
from the points of view of, voltage compensation, and harmonic suppression
capabilities are investigated. The system circuit parameters adopted are
presented in Table 5.1.
Table 5.1 System parameters of UPQC
System parameters
System frequency
Dc link capacitance
Dc-link voltage
Non-Linear Load
Shunt Inverter Filter
Series Inverter Filter
Switching Frequency
PWM Control
Coupling transformer
Specifications
50 Hz
C1=4400 F,C2=4400 F
600V
R =20 ohms,
L=15 mH,
2.6 KVA
L=5.5 mH ,
C=12 µF
L=5.5 mH ,
C=12 µF
9730 Hz
Hysteresis control
3.3 KVA
119
5.5.1
Compensation of Current Harmonics
Figure 5.5 (a) Load Current before compensation
Figure 5.5 (b) Source Current after compensation
Figure 5.5 (c)Reference Current of shunt Inverter
Figure 5.5(d) Capacitor voltage of Shunt inverter
The source is selected to be a highly harmonically distorted and
unbalanced. The waveform of load Current before compensation is shown in
Figure 5 . 5 ( a ) . When the source voltage is highly distorted by harmonics
which is cannot completely be removed by the series active filter, thus the
waveform of the load voltage is not completely sinusoidal. When the current
is compensated by taking the reference source current in which the q-axis
120
reference source current equals zero, and r-axis component is used to
compensate the neutral line current.
Load harmonics and reactive power required by the load is
compensated by injecting equal magnitude of harmonics but opposite
polarity. The hysteresis current controller is used for synthesizing the
compensating current with the current track band width of 0.018 ampere.
The reference current of shunt inverter consists of harmonic components of
non linear load, reactive current corresponding to load reactive power, power
losses due to inverters and DC voltage regulation current.
Using the reference load voltage wave and the q axis source
current, the source current can be compensated to sinusoidal as shown in
Figure 5 . 5 ( b ) . The simulation result verifies that if there is a balanced
sinusoidal reference, with the p-q-r theory the source current can be
compensated to be balanced and sinusoidal. The load absorbs an average
active power of approximately 2.6 kW.Figure 5.5 (c) shows reference Current
of shunt Inverter.
The THD of the distorted three-phase line currents (Ia, Ib & Ic ) are
36.47%, 31.42% and 38.27% respectively. The THD of Source current in
phase A, B and C has reduced to 4.23%, 4.97% and 4.56% respectively. It is
clear from that graph, the response time is less than 5 ms. DC link capacitor
voltage is almost kept constant during voltage unbalance . The results show
that a successful reduction in harmonics of the supply current is obtained. The
PI controlled dc-link capacitor voltage is nearly kept at 600 Vdc. Capacitor
voltage
of
Figure 5.5(d).
Shunt
inverter
under
non
linear
load
is
shown
in
121
5.5.2
Compensation of Load voltage
Load voltage with harmonics is shown in Figure 5.6(a). Since the
load voltage is compensated to be balanced and sinusoidal, the load voltage in
the p-q-r frames with respect to itself is a constant value. Also, the q and raxis of the source current is compensated to zero, with a dc value in p-axis,
this shows that the current is compensated to be balanced and sinusoidal.
Notice that the source voltage have no dc components in the q-axis and r-axis,
and since the current is now have a p-axis component only.
Figure 5.6(a) Load voltage with harmonics
Figure 5.6(b) UPQC voltage
Figure 5.6(c) Load voltage after compensation
122
This means that there is no phase difference between the
fundamental positive sequence of the source voltage and the load voltage, and
thus the power looping in the UPQC system has a minimum value.
Load voltage is compensated to be balanced and sinusoidal, thus
there is no influence to the load side even if there is a fault at the source side.
Therefore, series active filter needs more active power from the dc Link and
thus there is an increase in the power drawing from the dc storage. The
integration controller compensates this power with a transient process and the
drop of dc stops after the system reaches its steady state .As expected, the
steady-state dc voltage will be lower if the required power from the series
active filter is larger. The average active power supplied by the source is
approximately the same before and after the fault in steady state, except that
there may be a difference in the loss power. This shows that the power
looping in the UPQC does not consume extra power from the source.
The compensated voltage of series active filter is shown in Figure
5.6(b). Load voltage after compensation is shown in Figure 5.6(c). The Total
harmonic Distortion (THD) of the distorted three-phase load voltages are
51.87%, 48.25% and 49.56 % respectively. The THD of load voltages in
phase A, B and C has reduced to 5.23%, 5.11% and 5.05 % respectively. The
simulation result verifies, the load voltage can be compensated to be balanced
and sinusoidal.
5.6
CONCLUSION
A novel control strategy to generate the reference source current
and reference load voltage under distorted and unbalanced load and source
condition is presented in this chapter. The MATLAB/Simulink based
simulation results show that the distorted and unbalanced load currents and
123
load voltages seen from the utility side act as perfectly balanced source
currents and load voltages are free from distortion.
The drawbacks of the control scheme are uncompensated load
reactive power and failure of control during supply voltage unbalance. When
the load is unbalanced then the p and q powers are time variant quantities that
do not provide instantaneous information on power properties of the load.
Indeed, both powers are involved quantities, dependent at the same time on
two different power phenomena. An additional analysis is required to identify
the active, reactive, unbalanced and apparent powers after the p and q powers
are recorded over the entire period of their variability. Consequently, the IRP
p-q Theory, although it is based on a time-domain approach to power theory,
has no advantages over the frequency-domain approach with respect to the
time interval needed for identification of power properties of three-phase
loads.
According to the IRP p-q Theory, the instantaneous reactive current
can occur in linear circuits without reactive elements and consequently, when
the reactive power is zero. Moreover, the instantaneous active current can
occur when the active power is zero. These conclusions from the IRP p-q
Theory contradict the common meanings of the active and reactive currents.
Moreover, these two currents in systems with sinusoidal supply voltage and
without any loads that generate harmonics are non-sinusoidal. It means, the
IRP p-q Theory misinterprets power phenomena in electrical circuits.
Moreover, it does not reveal the load imbalance as the cause of power factor
degradation. It interprets the load imbalance as a loading that causes only a
change of the active and reactive currents.