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IEEE MELECON 2006, May 16-19, Benalmádena (Málaga), Spain
Different Approaches Assesment in Active Power
Filter Compensation
R. S. Herrera, P. Salmerón, S. P. Litrán, J. Prieto
Department of Electrical Engineering
Escuela Politécnica Superior, Universidad de Huelva
Huelva, Spain
[email protected], [email protected], [email protected], [email protected]
is carried out about its performance in the compensation.
Four formulations have been used, besides the original
one. They have generated interesting contributions
respect to the original theory. They are the Park
transformation or d-q coordinates, [3], [4], the modified
or cross product theory, [5], [6], the new proposal named
p-q-r theory, [7], [8], and the vectorial theory, [9]. At the
end, this paper corroborates that a more general
objective is possible: to get balanced and sinusoidal
source current for any voltage supply conditions.
Abstract—In this paper, the five main formulations of the
instantaneous reactive power theory have been chosen to
study a non linear load compensation. They are p-q
original theory, d-q transformation, modified or cross
product formulation, p-q-r reference frame and vectorial
theory. The obtention of the compensation current
according to each formulation has been established. Next,
the behavior of an Active Power Filter (APF) implemented
with those different control algorithms have been studied.
The APF control strategies have been implemented in an
experimental platform constituted by a 20 kVA power
inverter and a 400 MHz Digital Signal Processing (DSP)
controller board. Results got from an unbalanced and
non-sinusoidal three-phase four-wire system have been
compared. The final analysis shows that, in general, the
five theories present a different behavior, depending on
the supply voltage, respect to the distortion. However, all
of them widely decrease the waveforms distortion.
Moreover, a more general compensation objective is
possible. It obtains balanced and sinusoidal source current
in any conditions of supply voltage.
I.
II.
REACTIVE POWER THEORY
Table I presents the expressions of the currents
respect to the powers according to each strategy (first
column). Besides, the table presents the control strategy
derived from each theory (second column). They have
been calculated imposing the restrictions detailed by the
authors in the corresponding theory development.
In table I, voltage vector is:
ª u 1 º ª v 0 º ªe p º
& « » « » « »
u «u 2 » «vD » « 0 »
«¬u 3 »¼ «v E » «¬ 0 »¼
¬ ¼
Current vector is:
ª i º ª i º ª i º ªi º
& « 1» « 0 » « 0» « p»
i «i2 » «iD » «id » «iq »
«¬i3 »¼ «i E » «iq » «¬ ir »¼
¬ ¼ ¬ ¼
&
&
v0 is the zero sequence voltage vector and v
INTRODUCTION
At the beginning of the eighties, the named
instantaneous reactive power theory appeared. Its
objective was to find an effective control strategy to
compensate three-phase non-linear loads by mean of
APF. The original theory was named p-q formulation
and it was applied to three-phase three-wire non-linear
systems with a symmetrical constitution and a sinusoidal
voltage supply system, [1].
This is the formulation which has been applied in a
systematic way to make the three-phase non-lineal loads
compensation till nowadays. Nevertheless, at the end of
the eighties other formulations were proposed. A
comparative evaluation of some of those theories was
carried out when they were applied to get APFs control
algorithms for unbalanced three-phase three-wire
systems with non-sinusoidal voltage. At these
conditions, each theory produced different results,
without obtaining the opportunity to establish, in a
general way, the superiority of any theory over the
others, [2], [3]. In the nineties, the interest was specially
focussed on the study of three-phase four-wire systems
at most general conditions: unbalanced and non
sinusoidal source and non-linear unbalanced load.
Besides,
vDE
vD2 v E2
and
v 0DE
(1)
(2)
& &
u v0 .
v 02 vD2 v E2
About the power terms, p(t)=pDE(t) is the instantaneous
real power without zero sequence component, P is its
constant component and ~
p t its variable one. pu(t) is
the total instantaneous real power. Sub index ‘C’ refers
to the compensator, sub index ‘S’ to the source and sub
&
&
& &
index ‘L’ to the load. q t qDE t v u i is the
instantaneous imaginary power vector (and its
magnitude is the instantaneous imaginary power) in p-q
and vectorial theories. In modified p-q and p-q-r
theories it is as follows:
ª q0 º
& « » ªqr º
(3)
q « qD » « »
q
«q E » ¬ q ¼
¬ ¼
In this paper, strategies got from five instantaneous
power theory formulations are analysed and applied to
unbalanced and non-linear systems. A comparative study
1-4244-0088-0/06/$20.00 ©2006 IEEE
BACKGROUND ABOUT INSTANTANEOUS
1090
TABLE I. COMPENSATION CURRENTS.
APLC Currents
Currents respect to powers
ª i0 º
« »
«iD »
«i E »
¬ ¼
p-q
ª i0 º
« »
«iD »
«i E »
¬ ¼
p-q Modified
ª i1 º
«i »
« 2»
«¬i3 »¼
0
v 0 vD
v0 v E
0
vD
vE
1
2
v 0 vDE
2
ªvDE
«
« 0
« 0
¬
0 º ª i0 º
»« »
v E » «iD »
vD »¼ «¬i E »¼
0
vE
vE
0
vD
v0
0
v 0 vD
v0 v E
ªiC 0 º
« »
«iCd »
«iCq »
¬ ¼
ª pu º
vD º « »
» q0
v0 » « »
« qD »
0 »¼ « »
¬«q E ¼»
ª1 0 0 º ª p u º
1 «
« »
0 1 0 »» « q r »
«
ep
«¬0 0 1»¼ « q q »
¬ ¼
ª v1 º
1 p0
p « »
v2 v2 « »
3 v02
«¬v3 »¼
ª iC 0 º
« »
«iCD »
«iCE »
¬ ¼
º ª p0 º
»«
»
v 0 v E » « pDE »
v 0 vD » «¬ qDE »¼
¼
0
ªvDE
1 «
0
vDE ««
¬ 0
ª v0
1 «
«vD
v 02DE «
¬v E
ªi p º
«i »
« q»
«¬ i r »¼
p-q-r
III.
v 0 vD2 E
ª i0 º
« »
«i d »
«i q »
¬ ¼
d-q
Vectorial
1
2
ªvDE
«
« 0
« 0
¬
ª iC 0 º
1
« »
«iCD »
2
«iCE » v 0DE
¬ ¼
ª iC 0 º
« »
«i CD »
«i CE »
¬ ¼
ªv 2 v 3 º
ªv 0 º
«v » 1 q « v v »
3
1»
« 0»
3 v 2 ««
«¬v0 »¼
v
v
¬ 1 2 »¼
A MORE GENERAL COMPENSATION OBJECTIVE
Different objectives have been considered to define the
concept of three-phase loads compensation with APFs,.
Mainly, two of them are presented: constant power
compensation and unity power factor compensation. In
most of the cases constant power compensation has been
applied with the direct application of the p-q formulation.
In constant power compensation, the target is that the
source supplies a constant instantaneous power after
compensation, equal to the load active power, [10]. On the
other hand, unity power factor compensation is used to
achieve that the source current has the same distortion and
symmetry conditions as the source voltage. It means that
the source current is collinear to the supply voltage. In this
situation, the source supplies the load active power, but the
instantaneous real power is not constant after
compensation.
When source voltage is sinusoidal and balanced,
constant power and unity power factor compensation
results are exactly the same. In fact, in both cases, source
voltage and current after compensation are balanced and
sinusoidal, and the instantaneous power supplied by the
source is constant. Taking it into account, the definition of
the ideal reference conditions, finding the compensation
optimal objective, is necessary.
1091
1
v 0DE
ª
«
« v0
«
« vD
«
«
«v E
«¬
ª v0
«
«vD
«v E
¬
0
v 0DE v E
vDE
v 0DE vD
vDE
ª i C1 º
«i »
« C2 »
¬«iC 3 ¼»
º ª p L0 º
»
p L PL 0 »»
v 0 v E » «« ~
v 0 vD » «¬ q L »¼
¼
0
ªi L 0 º
«~ »
« iLd »
«i Lq »
¬ ¼
0
vE
vE
0
vD
v0
p Lu º
ª~
vD º «
»
» « q L0 »
v0 »
« q LD »
0 »¼ «
»
¬« q LE ¼»
~
ºª
p Lu
»«
vDE » «
vp
v 0 vD » «
q Lr
»«
vDE » «
vp
v 0 v E » « q Lq v 0
p
»«
val ( Lu
vDE »¼ «¬ v p
vDE
vp
ª i L1 º
«i » P Lu
« L2 » V 2
¬«i L 3 ¼»
º
»
»
»
»
»
»
)»
»¼
ª v1 º
«v »
« 2»
¬«v 3 ¼»
Due to the fact that the electrical utilities produce, in
general, the electrical power as sinusoidal and positive
sequence phase voltages, this has been established as
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. Anything which produces a no conformity
respect to these reference conditions, supposes a poor
electric power quality. Moreover, all the compensation
devices must achieve a sinusoidal and balanced source
current, in phase with voltage positive sequence, with any
supply voltage condition and with all kinds of load. These
requirements involve the ideal reference conditions for
supply current and they are the currents that define the
compensation objectives more used recently, [11]-[13].
Section II established the control strategy derived from
the vectorial theory. This strategy is equivalent to the unity
power factor compensation objectives eliminating the
neutral current. If voltage supply is balanced and
sinusoidal, the source current is balanced and sinusoidal,
too. Nevertheless, if the voltage supply is unbalanced and
non sinusoidal, the source current is unbalanced and non
sinusoidal, too. In this case, to obtain source current
balanced and sinusoidal, it is necessary to modify the
control strategy as follows:
&
iC
where
&
u1
& P &
i Lu
u1
U 1 2
represents
the
(4)
positive
sequence
fundamental wave of the three-phase voltage and U 12 the
square of the positive sequence fundamental wave of
voltage RMS value.
IV.
EXPERIMENTAL RESULTS
In this work, the APF control has been implemented by
mean of a specific digital signal processor (DSP) board
developed by dSPACE, [14]. In particular, DS1103 peer to
peer connection (PPC) controller board is equipped with a
Power PC processor for fast floating point calculation at
400 MHz. This hardware allows to program the control
circuit via Simulink. In this way, all the control circuit
components are configured graphically within the
Simulink. The RTI translates the Simulink model to C
language, it generates the real-time executable program,
and it downloads it in the controller board. To check the
approach proposed in this paper, it was applied in a threephase four-wire unbalanced ac-regulator compensation.
The voltage source is unbalanced. Its phase rms values
are 203.84, 147.81 and 221.92 V, respectively. The load is
composed of three regulators with a serial inductive load in
each phase connected in star (a resistor of 52.2:/phase and
an inductance of 150 mH/phase). The software running in
the real-time processor carries out the control. It calculates
the reference source current as it was presented in section
II and III. The difference between the real compensation
current and the calculated before is the input to the PWM
module. Its output are the power circuit IGBTs trigger
signals.
Table II presents the results of applying the six control
strategies to the experimental prototype presented. First
column shows the phase one of voltage and source current
First row presents the waveforms before compensation.
The load current presents a displacement respect to the
voltage. As can be seen in rows 2 to 7, it is eliminated by
applying the six strategies. The second column shows the
three-phase
source
current
waveforms.
Before
compensation (second column first row), it is unbalanced
and distorted. The distortion is due to the kind of load and
the unbalance is due to the voltage unbalance. The other
rows present the three-phase waveforms after
compensation using each control strategy. All of them
show a much lower distortion than the presented by the
load current. Nevertheless, only the global control strategy,
presented in section III, achieves to get balanced and
sinusoidal source current after compensation, table II last
row.
wave. At this conditions: a) All the formulations achieve
the target if supply voltage is balanced and sinusoidal. b)
None of them achieve, in their original development, a
zero distortion index value if supply voltage is non
sinusoidal. c) Control strategies derived from the p-q, d-q,
p-q-r and vectorial theories allow to obtain control
algorithms in cases 2 and 3 with a distortion below the
10%. Modified p-q gets higher values. d) P-q, modified pq, p-q-r and vectorial formulations suppose a null
compensator average power and d-q requires a
compensator average power not null; p-q, p-q-r, d-q and
vectorial formulations get a null neutral current but
modified p-q does not eliminate the neutral current.
A more general compensation objective is possible. It
obtains balanced and sinusoidal source current in any
conditions of supply voltage, table II last row.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
V.
CONCLUSSIONS
In this paper, the most relevant formulations of the
instantaneous reactive power theory have analysed. The
control strategies derived from each formulation presented
have been compared. The compensation target considered
in the comparative has been the obtention of sinusoidal
source current in phase with the symmetrical positivesequence component of the supply voltage fundamental
1092
[13]
[14]
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TABLE II. SOURCE CURRENT WAVEFORMS AFTER COMPENSATION APPLYING THE DIFFERENT CONTROL STRATEGIES
Phase one of voltage and source current
Before compensation
After compensation
applying p-q theory
After compensation
applying d-q theory
After compensation
applying modified pq theory
After compensation
applying p-q-r theory
After compensation
applying vectorial
theory
After compensation
applying the global
control strategy
1093
Three-phase source current waveform