IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 3, JULY 2006
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Distorted and Unbalanced Systems Compensation
Within Instantaneous Reactive Power Framework
Patricio Salmerón and Reyes S. Herrera
Abstract— – theory has been widely used to control active
power filters since its formulation in 1983. The compensation
strategy used by the – theory has not suffered modification; so
it used the constant power compensation strategy. In this way, the
supply instantaneous power after compensation is constant. This
kind of compensation strategy presents good results in the case of
balanced and sinusoidal voltages; however, it is not appropriate
in the case of unbalanced or nonsinusoidal voltages. This paper
extends the – theory to get control strategies which allow to
attack the unity power factor compensation or the balanced and
sinusoidal currents compensation.
Index Terms—Active power filters, harmonics, instantaneous reactive power, power quality, three-phase systems.
I. INTRODUCTION
T
HE integration of power electronics in the commercial and
industrial processes has caused a considerable increase of
nonlinear currents. The problems associated with these distorted
currents have made harmonics compensation a priority task.
There are several ways to achieve the target of nonlinear currents compensation, and one of the currently most used is the active power filter (APF) [1]–[6]. This is a power-electronic converter which supplies the reactive current required by the load.
Imposing a compensation target to the control circuit, it calculates the current supplied by the compensator according to the
instantaneous reactive power theory or other formulations later
proposed [7]–[16].
In this paper, the different control strategies proposed within
instantaneous reactive power theory have been applied to a
three-phase four-wire system similar to that presented in Fig. 1.
It is a three-phase four-wire system composed of an unbalanced
and nonsinusoidal voltage source that supplies a nonlinear
unbalanced load. This load involves an ac regulator composed
of bidirectional thyristors and series impedances with different
values in each leg to obtain unbalanced conditions. The compensator injects the reactive current required by the load. Within
the paper, lower case represents instantaneous values, upper
case average values, subindex “L” is the load requirement,
subindex “C” is the compensator supply, and subindex “S” is
the source supply.
From the time that the instantaneous reactive power theory
appeared, different formulations were generated during the last
Manuscript received September 21, 2004; revised November 8, 2005. Paper
no. TPWRD-00444-2004.
The authors are with the Electrical Engineering Department,
Huelva University, Huelva 21819, Spain (e-mail: patricio@uhu.es;
reyes.sanchez@die.uhu.es).
Digital Object Identifier 10.1109/TPWRD.2006.874115
Fig. 1. Nonlinear and unbalanced three-phase load static compensation.
Fig. 2.
Matlab simulink implementation diagram.
decade; this work is focused on the first of them, known as –
axes or original instantaneous reactive power theory.
Fig. 2 presents a Matlab-Simulink implementation diagram.
(It can be seen as the source is composed of three single-phase
sources and a serial resistor in star configuration). On the other
hand, the load is composed of two back-to-back silicon-controlled rectifiers (SCRs) and a serial resistor whose value is different in each phase. Thus, a nonlinear load in star connection
is obtained. The block identified as “Control Table I” is a mask
0885-8977/$20.00 © 2006 IEEE
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 3, JULY 2006
TABLE I
CONTROL STRATEGIES
whose content is different for each specified control strategy. Its
target is to obtain such compensation currents defined in Table I.
II. CONSTANT POWER COMPENSATION
The instantaneous reactive power theory was formulated at
the beginning of the 1980s [7] and this is the formulation which
has suffered the largest diffusion along these years. That is the
reason to be the most used as control strategy for the APF. The
– theory was developed for three-phase three-wire systems
with balanced and sinusoidal source voltages. The compensation target assumed was getting a constant instantaneous source
power.
This theory is based on a coordinates transformation from the
system (Fig. 3).
phase reference system (123) to the
The transformation matrix is associated as follows:
Fig. 3.
0 reference system.
(2)
Looking at these equations, it can be deduced that
(3)
The different power terms are defined as follows:
(1)
(4)
SALMERON AND HERRERA: DISTORTED AND UNBALANCED SYSTEMS COMPENSATION
where
is the zero-sequence (real) instantaneous power,
is the
real instantaneous power, and
is the imaginary instantaneous power. Considering the [T] inverse matrix, it is possible to calculate the current components by means of the different power terms. The expression is shown in the next equation
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.
where
The strategy assumed by this theory since its origin has been
the obtention of a constant instantaneous power in the source
side with the only restriction of getting a null average instantaneous power exchanged by the compensator .
In this way, the first row in Table I shows the control strategy
derived from this theory as its original formulation. In fact, to
calculate the compensator current, according to Fig. 1 references, it is verified that
the same distortion and symmetry conditions as source voltages. It means that the source currents are collinear to the supply
voltages. In this situation, the source supplies the load active
power, but the instantaneous real power is not constant after
compensation [8].
When the source voltages are sinusoidal and balanced, constant power and unity power factor compensations results are
the same. In fact, for sinusoidal and balanced source voltages,
if compensation currents cancel reactive, distortion, and unbalanced current components, source currents will be sinusoidal
and balanced in the same way as voltage. And for that reason,
instantaneous power supplied by the source will be constant.
However, when the supply voltage is unbalanced and nonsinusoidal, the situation will not be the same.
Table I, in the third row, shows the control strategy based on
the – theory, which gets unity power factor in the source side
after compensation. Up to now, this strategy adoption in the –
theory frame has not been planned. In this new condition, the
source current must be proportional to the voltage supplied
(6)
(12)
where
is the total active power incoming to the load. Equation (6) can be expressed in the following way:
The compensator instantaneous power is the difference between the total real instantaneous power required by the load
and the part supplied by the source as explained above.
To satisfy that the active power exchanged by the compensator is null, the proportionality constant must be fixed to the
next value
(5)
(7)
(8)
which, effectively, fulfills that the average value
,
represents the ac part of
.
where
On the other hand, the instantaneous imaginary power exchanged by the compensator must be the same as the instantaneous imaginary power required by the load
(9)
The second column in the table informs whether the theory
eliminates the neutral current and the third one informs about
the active power supplied by the source after compensation,
showing whether it is constant in time. The strategy shown in
the first row does not eliminate the neutral current but the active
power exchanged by the compensator is null.
It is possible to seek a strategy that eliminates the neutral current. To impose this new restriction, in the – theory frame, a
simple modification is only necessary: the zero-sequence power
used to calculate the compensator current will be the zero-sequence phase power required by the load [3]
(10)
(11)
The second column shows that, in fact, this new strategy eliminates the neutral current. Moreover, the instantaneous power
supplied by the source goes on being constant.
(13)
thus, since
, it follows that:
(14)
where
is the square of voltage vector rms value
(15)
Finally, after compensation
(16)
From (16) and with the target of filling Table I third row, and
taking into account that the compensator instantaneous power
is the difference between the total real instantaneous power required by the load and the part supplied by the source, it follows:
(17)
(18)
Besides, for the instantaneous imaginary power, it is verified
(19)
III. UNITY POWER FACTOR COMPENSATION
The unity power factor compensation is a strategy widely
used along the time. The target is to get source currents with
Table I presents the compensator currents equations in
coordinates.
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This control strategy does not permit to eliminate the neutral
current. To satisfy this restriction, the strategy must be changed
to the form shown in Table I, fourth row. In this case, the power
factor is a little lower than the unity. Nevertheless, the neutral
current is eliminated even in the case that voltages are not balanced or sinusoidal.
IV. SINUSOIDAL AND BALANCED
SOURCE-CURRENT COMPENSATION
Fig. 4. Method to obtain the voltage fundamental component.
For the last years, both compensation objectives presented
in this paper have not been enough to solve the new problems
associated with the voltage and current distortion. The definition
of a new compensation target based on an ideal reference case
is necessary.
Due to the fact that the electrical companies produce, generally, the electrical power as sinusoidal and balanced voltages,
it has been a reference condition in the supply. A resistive
load—balanced and linear—is considered as the ideal reference
load. A reference voltage applied to an ideal reference load will
generate sinusoidal and balanced currents, in phase with voltages [12]–[14]. Anything which produces a no conformity with
these reference conditions assumes poor electric power quality.
Compensation devices must achieve that the source current is
sinusoidal, balanced, on positive sequence, and in phase with
voltages positive-sequence fundamental component. It must
be fulfilled for any supply voltages conditions and load kinds.
These requirements involve the ideal reference conditions for
the supply current and they are the currents that define the last
compensation objectives assumed in this paper.
If voltages are balanced and sinusoidal, ideal currents will be
(as in earlier strategy) proportional to the voltages. The proportionality constant must have the value established in (12)–(15)
too.
If voltages are balanced nonsinusoidal, the ideal reference
current must be proportional to its fundamental component.
The proportionality constant is fixed to satisfy the restriction of
having null active power exchanged by the compensator. The
ideal current expression is as follows [16]:
(20)
where
and
represent the fundamental components of the
, voltage, respectively, and
the square of the voltage
fundamental component root mean square (rms) value.
If voltages are unbalanced nonsinusoidal, the source current
expression becomes as shown in (21)
(21)
and
represent voltage positive sequence fundawhere
mental components in
coordinates and
is the square
positive-sequence fundamental component
of the voltage
rms value.
is proportional to the voltage positive-sequence
In (21),
fundamental component. The proportionality constant value is
fixed to satisfy the restriction of null active power exchanged by
the compensator.
In Table I, the last row shows the compensation equations to
get balanced and sinusoidal supply currents. They are, as earlier
coordinates. In this case, it must be noted that
strategies, in
at first time, it is necessary to impose restrictions not only to the
and
, even besides to the
instantaneous real powers
. This new restriction is
instantaneous imaginary power
necessary to get sinusoidal and balanced source currents even if
voltages are nonsinusoidal or unbalanced.
Moreover, this strategy achieves to eliminate the neutral current with a null active power exchanged by the compensator.
V. SIMULATION RESULTS
The five strategies presented in previous items and shown in
Table I have been applied to the simulation platform shown in
Fig. 1.
The method used to obtain the voltage fundamental component necessary in the last strategy is based on the use of lowpass
filters, multipliers and adders as it is shown in Fig. 4. In this
figure, for any signal which presents the next form
(22)
the product composed by the voltage component
and
has a constant term
being
the waveform
the voltage fundamental component amplitude and
the
voltage fundamental component phase.
On the other hand, the product composed by the same voltage
component
and the waveform
has a constant term
. Multiplying both constant terms per 2 and per
and
, respectively, as can be seen in Fig. 4, the phase
1 voltage fundamental component waveform is built.
The results from the different simulation strategies are presented in Figs. 5–8 which show two periods (0.04 s).
Fig. 5 presents the voltages supply correspondent to each
case. The first graph shows a balanced and sinusoidal voltage
system, the second one shows an unbalanced and sinusoidal
voltage system, and the last one shows a balanced and nonsinusoidal voltage system.
Fig. 6 presents the current waveforms corresponding to
case 1, balanced and sinusoidal supply voltage from each
control strategy, before and after compensation. Fig. 7 presents
the current waveforms which are obtained by applying an unbalanced and sinusoidal voltage system, that is the case 2. And,
finally, Fig. 8 presents the case 3, balanced and nonsinusoidal
supply voltage.
SALMERON AND HERRERA: DISTORTED AND UNBALANCED SYSTEMS COMPENSATION
Fig. 5.
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Supply voltages (V).
In Figs. 6–8 first graph, it can be seen that the three-phase
source current before compensation (i.e., the three-phase current required by the load) is a strongly nonlinear and unbalanced current even if the voltage supply system is balanced and
sinusoidal.
Of course, the load currents present a different distortion and
unbalanced behavior for each supply condition corresponding
to each case.
Case 1: Balanced and Sinusoidal Voltage Supply: Fig. 6, in
the first graph, presents the three load phase currents. The three
waveforms’ different peak values allow to observe the load unbalance effect.
The other five graphs correspond to the source-phase currents
after compensation applying such compensation strategies included in Table I. So these strategies get constant source power,
unity power factor, and balanced and sinusoidal current. In this
case of the balanced and sinusoidal source, all of the strategies
fit the results and they present balanced and sinusoidal source
currents.
Fig. 6. Three-phase current (A) obtained by applying a balanced and
sinusoidal voltages system.
Case 2: Unbalanced and Sinusoidal Voltage Supply: Fig. 7,
in the first graph, presents the load currents. In this case, currents
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Fig. 7. Three-phase current (A) obtained by applying an unbalanced and
sinusoidal voltages system.
are unbalanced due to the load unbalance effect and to the supply
voltage asymmetry, simultaneously.
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 3, JULY 2006
The second graph presents the three source-phase currents
after compensation when the strategy named constant power
without eliminating the neutral current is applied. This graph
shows a strong distortion in the three-phase waveform. Nevertheless, the third graph shows that with the only additional
restriction of eliminating the neutral current in the same strategy
(constant power compensation), the current distortion is widely
reduced. This second strategy does not obtain sinusoidal source
current although both strategies get constant instantaneous
power supplied by the source after compensation.
The fourth graph presents the source current after compensation according to the third strategy shown in Table I, third row.
This is the unity power factor strategy without taking into account the neutral current elimination. In this case, the source
current waveform is collinear to the voltage and, although this
strategy compensates the unbalance load, it is not able to compensate the supply voltage zero-sequence phase component. So
this strategy does not eliminate the neutral current. However,
imposing the additional restriction of eliminating the neutral
current, voltage and current waveforms are not collinear. This
strategy is shown in Table I, fourth row. Nevertheless, the unbalance is low in this case too. Neither of these strategies will
get constant instantaneous power supplied by the source after
compensation if the voltage supply is unbalanced. The power
factor is the unity applying the first of these two strategies and
it is almost the unity in the other one.
The three-phase current waveform shown in last graph is the
only one which is balanced and sinusoidal. It corresponds to the
strategy presented in Table I, last row, that imposes getting sinusoidal and balanced source currents in phase with the voltages
and carrying the active power required by the load.
Case 3: Balanced and Nonsinusoidal Voltage Supply: In
Fig. 8, first graph, the three-phase source current before compensation can be seen (i.e., the three-phase current required
by the load with voltage distortion). The current is strongly
distorted due to the simultaneous action of the load nonlinearity
and the applied voltage distortion.
As in the two earlier cases, in Fig. 8, in the second to sixth
graphs, source currents corresponding to the different compensation strategies are presented. The constant power without
eliminating neutral current strategy (second graph) presents the
higher distortion. This distortion decreases if the condition of
eliminating the neutral current is included (third graph).
The fourth and fifth graphs present the results from applying
the strategies named unity power factor, where the source currents reproduce the voltages distortion. In the case of not eliminating the neutral current, the power factor is one.
However, if the condition of eliminating the neutral current is
included, the phase currents cancel the three multiple harmonics
at the expenseof apower factor that is slightly smaller that the unit.
At last, the sixth graph presents the results from the sinusoidal
source currents strategy. The source currents are now balanced
and sinusoidal. The load unbalanced and the source voltage distortion have been compensated.
In summary, a strongly nonlinear and unbalanced load has
been considered to be studied in three different cases of supply
voltages. In case 1, the load unbalanced effect is completely
SALMERON AND HERRERA: DISTORTED AND UNBALANCED SYSTEMS COMPENSATION
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Fig. 9. Dynamic response. Current in amperes.
TABLE II
APF VA VALUES
Fig. 8. Three-phase current (A) obtained by applying a balanced and
sinusoidal voltages system.
compensated by all of the control strategies and they fit the results. In case 2, both effects of load and source voltage unbalance are simultaneously present. The only strategy which gets to
compensate the combined effect is the correspondent to the sinusoidal source current. The strategies that do not achieve to eliminate the neutral current are the constant source power without
eliminating the neutral current and the unity power factor. This
last strategy does not compensate the voltage zero-sequencephase component.
To show the system dynamic behavior, the phase 2 serial
resistor value has been scaled by a factor of 5. Fig. 9 presents
two graphs: the first one shows the phase 2 load current and the
second one shows the three-phase source current. This figure
shows that the system follows the load change in only one
period.
To finish the results, Table II presents the APF VA values in
each case presented in the paper and for each strategy studied.
The values are very similar. The only strategy which needs a
higher-rated apparent power by the compensator is defined in
Table I, in the first row: constant source power without eliminating neutral current in the case of unbalanced voltage or nonsinusoidal.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 3, JULY 2006
VI. CONCLUSION
An exhaustive analysis of the – theory instantaneous reactive power theory has been presented in this paper. The three
main compensation objectives have been presented and studied
to generate five control strategies based on the – theory instantaneous reactive power theory, one corresponding to each
compensation target. Thus, a new analysis has been carried out
where it is shown that with few modifications to the control
strategy developed by the authors of the original theory, any
compensation target imposed upon the system may be achieved.
So, the original – theory does not present any problem to be
used as the base of an active power filter control algorithm, with
any voltage conditions in the point of common coupling (PCC).
The results have been corroborated by the correspondent simulations in the Matlab-Simulink environment.
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Patricio Salmerón was born in Huelva, Spain. He
received the Ph.D. degree in physics from the University of Seville, Seville, Spain, in 1983.
From 1983 to 1993, he was with the Department
of Electrical Engineering, University of Seville,
and since 1993, he has been Professor of Electric
Circuits and Power Electronics with the Electrical
Engineering Department, Huelva University, Huelva.
Currently, he is the Head of the Escuela Politécnica
Superior. He has joined various projects in connection with research in power theory of nonsinusoidal
systems and power control in electrical systems. His research includes electrical
power theory, active power filters, and artificial neural networks.
Reyes S. Herrera was born in Huelva, Spain. She
received the Industrial Engineering degree from the
University of Sevilla, Sevilla, Spain, in 1995.
From 1996 to 2001, she was with the Atlantic
Copper Smelter, Huelva. Currently, she is a Lecturer
of Electrotechnical and Power Electronics with
the Electrical Engineering Department, Huelva
University, Huelva. Her research includes electrical
power theory and active power filters.