APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
Application of the Instantaneous Power Theories in Load Compensation
with Active Power Filters
Patricio Salmerón Revuelta, M. Reyes Sánchez Herrera
Universidad de Huelva. Dpto. Ingeniería Eléctrica y Térmica.
Ctra. Palos de la Frontera s/n. Palos de la Frontera (21819).
patricio@uhu.es, Reyes.Sanchez@die.uhu.es
Key words
Active filters, Harmonics, Power Quality, Three phase systems.
Abstract
Today, the main theory for the design of APF (Active Power Filter) control for non-linear three-phase
loads compensation has been the instantaneous reactive power theory. Since 1983 when p-q original
theory appeared, there have been proposed many different formulations applied to the APF control.
However, the compensation strategies derived from the different instantaneous reactive power
formulations present a different behaviour when unbalanced and non-sinusoidal source voltages are
applied to three-phase four-wire systems.
In this paper, the behaviour of different APF control algorithms got from the five most relevant
instantaneous reactive power formulations, according to the outstanding publication, is analysed. They
are p-q original, d-q transformation, modified or cross product formulation, p-q-r reference frame and
vectorial formulation. A platform of simulation with control + APF + load to test the different
algorithms has been built. The results got in the following three cases were compared: balanced and
sinusoidal, unbalanced and sinusoidal, and balanced and non sinusoidal source voltage. The analysis
of results from the five formulations proved that only the vectorial one allows to get balanced and
sinusoidal currents after compensation.
1. Introduction
Twenty five years ago the named instantaneous reactive power theory appeared with the objective of
finding an effective control strategy to compensate three-phase non-linear loads by mean of active
power filters APF, or its more correct denomination, APLC (Active Power Line Conditioner). The
original theory was the p-q formulation and it was applied to three-phase three-wire non-linear
systems with a symmetrical constitution and an ideal supply of sinusoidal voltages, [1]. The control
strategy got from the p-q formulation resulted efficient in the target proposed: sinusoidal source
currents after compensation with the same characteristics than the voltages. This is the formulation
which has been applied in a systematic way to approach the three-phase non-linear loads
compensation till nowadays. Nevertheless, at the end of the eighties other formulations were proposed
with identical results than p-q one in balanced non-linear systems with sinusoidal voltages [2-6]. A
comparative evaluation of some of those formulations was carried out when they were applied to get
APLCs control algorithms for unbalanced three-phase systems with non-sinusoidal voltage. At these
conditions, such formulation produces different results, without obtaining the opportunity to establish
in a general way the superiority of any of them over the others, [7-8]. After that, in the nineties, the
interest was specially focus on the study of three-phase four-wire systems at the most general
conditions: unbalanced and non sinusoidal source and non-linear unbalanced load. A basic objective
was to find control strategies which allow to cancel the neutral current with a null average power
transferred by the compensator. Nevertheless, there is not an adequate solution to get APLCs control
algorithms based on the mentioned formulations when it is being treated the distorted and unbalanced
systems compensation.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.1
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
In this paper, it is being analysed the strategies got from five instantaneous power theory formulations
applied to unbalanced and non-linear systems: the original p-q, [1], the Park transformation or d-q
coordinates, [7], the modified or cross product, [3-6], the new proposition named p-q-r, [9-10], and the
vectorial, [11]. It is being carried out a comparative study about its performance in the compensation
by mean of two indexes measured in the source currents after compensation: the THD value and the
neutral current rms value. Taking into account the five formulations, only the vectorial one allows to
get a compensation strategy which makes the source current sinusoidal and balanced (null THD) in
phase with the positive-sequence voltage source (null reactive power); besides it makes null the
neutral current. This control strategy is designed to maintain null the average power transferred by the
compensator. In this work, they have been used the mentioned five formulations in three operational
conditions: supply of an unbalanced and non-linear load in a three-phase four-wire system with 1)
balanced sinusoidal, 2) unbalanced sinusoidal and 3) balanced non-sinusoidal voltage.
Table I. Five Formulations on the Instantaneous Reactive Power Theory.
Voltages
p-q
Modified
p-q-r



 v0 
2

vα  =
v 
3
 β



1



 v0 
2

v  =
α
v 
3
 β



1
2
1
0
2
1
0
1
2
1
2
3
2
−
1
2
1
−
2
3
2
u p  v0αβ 
u  =  0 
 q  0 
ur  

v0αβ = v02 + vα2 + vβ2
Vectorial
1 

2  u
1   1
u 2 
−
2  u 
3
3  
− 
2 
1 

2  u
1   1
u 2 
−
2  u 
3
3  
− 
2 
 i0 
i  = 2
iα 
3
β
 i0 
i  = 2
iα 
3
β


 v0
i p 

1
i  =
 0
 q  v 0αβ 
i
r
 

vαβ

v0 
v1  u1   3
u1 
v
u = u2  ; v = v2  = u2  −  0 
3

v3  u3  v
u3 

0
 3
Powers

 1
1
1


2
2

 2
[T ]= 2 cosϑ cosθ − 2π  cosθ + 2π 
3
3
3




 2π 
 2π  
 0
sinθ −  sinθ + 
3  
3



i0 
i1 
i  = [ T ] i 
d
2
i 
i 
 3
 q
d-q
p-q
Currents
















−
1
1 

2  i
1   1
i2 
−
2  i 
3
3  
− 
2 
1 

2  i
1   1
i2 
−
2  i 
3
3  
− 
2 
1
2
2
1
2
3
2
1
−
1
0
1
2
2
1
−
2
3
2
1
0
vα
v 0α β v β
vαβ
v 0 vα
−
vαβ
v0
 p0 
 p = 2 0
3
q
 
 0
 pu   v0
 q0   0
q  =  v
 α  β
qβ  − vα


vβ

v 0α β vα   v 0 
−
  vα 
vαβ  v 
 β
v0 v β 
−

vαβ 
 pu 
q  =v
qr  p
 q
0
vα
−vβ
vα
− vβ
0
v0
0   i0 
v β  iα 
 
vα  i β 
vβ 
i 
vα   0 
 iα
− v0   
iβ
0   
1 0 0  i p 
0 1 0   i q 
0 0 −1  i 
 r
vαβ = vα2 + vβ2
 i1 
i = i 2 
i 3 
pu (t ) = u.i
p0 (t ) = v 0 .i
p (t ) = pu (t ) − p0 (t ) = v.i
2. Different Approaches about Instantaneous Reactive Power Theories.
In three phase systems, instantaneous phase voltages and currents can be transformed to another
reference frame by mean of a mathematical transformation represented by an orthogonal matrix, T.
When the matrix T is defined as the included within the Table I, with the arbitrary function θ(t)=ωt
where ω is the voltage angular frequency, T is the Park transformation. The application of Park
transformation to three generic three-phase quantities, gives their components in 0-d-q coordinates.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.2
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
If θ(t) is considered as θ(t)=0, then, it obtains the Clarke-Concordia components on the 0, α, β
reference frame. These last are the coordinates used by the original and modified formulations. P-q-r
makes up a further rotating reference frame while only the vectorial uses phase coordinates (fourth and
fifth row in table I).
Table I shows the voltages and currents space vectors for the five formulations. Besides, they include
the power terms. In a three-phase four-wire system, the real instantaneous power is defined as:
p u (t ) = u1 i1 + u 2 i 2 + u 3 i3 = vα iα + v β i β + v0 i0 = p (t ) + p 0 (t )
(1)
In the instantaneous reactive power theory, it is usual to define two instantaneous real power, p0(t) and
p(t), and other power variables as the instantaneous imaginary power, q(t), or instantaneous reactive
power components, q0(t), qα (t), qβ(t), table I.
Table II presents the instantaneous currents defined through the power variables. It is very important
to recall that, in the instantaneous power theory domain, an instantaneous active current and an
instantaneous reactive current appear. These currents are different from Fryze´s active and reactive
currents.
Table II. Instantaneous Currents from the Power Terms.
p-q
Instantaneous current components

 2
 
0
0   p0 
v αβ
 i0 
1
 0
i  =
−v v   p 
v v
α
0 α
0 β  
  v v 2αβ 
q
i

0
0
v
v
v
v  
 β 
0 β
0 α 

p-q
Modified
 i0 
i  = 1
iα  v 2 0αβ
β
p-q-r
i p 
i  = 1
 q  vp
 ir 
Vectorial
 v0
v
 α
v β
0
− vβ
vα
vβ
0
− v0
p
− vα   u 
q
v0   0 
q
 α
0  q 
 β 
1 0 0   pu 
0 1 0   q r 
0 0 −1  q q 
 
 i1 
 v1 
v 0 
v 2 − v3 
i  = p v  + 1 p0 v  + 1 q  v − v 
0
3
1
 2  v2  2 
3 v02  
3 v2 
i3 
v3 
v0 
 v1 − v 2 
From each formulation, control strategies to compensate a non-linear three-phase load by mean of an
APLC are derived in Table III. Compensation currents equations that are obtained after the
correspondent constrains imposed to the power terms are included. In p-q and p-q modified
formulations a constant source power eliminating neutral line current as control strategy is adopted,
[6]. The method called Reference Power Control eliminating neutral line current is used from p-q-r
reference frame, [10]. In d-q transformation, reference current control is considered, [7]. Finally, a
source current with a time average equal to the load power collinear with a reference voltage
(fundamental positive-sequence) is considered by vectorial formulation. With regard to the notation in
table III, for example, pL(t) is a load instantaneous real power, PL is the pL(t) time-average value. The
previous magnitudes with the symbol ~ represent the waveform ac component.
Table III third column includes the zero-sequence current; all the formulations attain a null neutral
current except the modified one. The fourth column includes the instantaneous power time average of
the compensator, that is null in all the formulations except in the d-q one.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.3
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
In this last case the compensator would need an external power source to obtain the compensation
current.
Table III. Instantaneous Compensation Currents.
APLC Currents
p-q
 iC 0 
1
 
iCα  =
2
iCβ  v0 vαβ
 
2
vαβ

 0
 0

p-q-r
Compensator
Power
0
PC = 0
0
PC ≠ 0
  p L0 

− v0 v β   ~
p L − PL 0 
v0 vα   q L 

0
v0 vα
v0 v β
iC 0  i L 0 
  ~ 
iCd  =  iLd 
iCq   ~

   iLq 
d-q
p-q
Modified
0
Zero Comp
Current
 iC 0 
1
 
iCα  = 2
iCβ  v0αβ
 


 v0
iC 0 

i  = 1 v
Cα
i  v 0αβ  α
 Cβ 

v β

 v0

 vα
v β

0
vβ
− vβ
vα
0
− v0
0
−
v 0αβ v β
vαβ
v 0αβ vα
vαβ
p Lu 
~
− vα  

  q L0 
v0 
q Lα 
0  

 q Lβ 
is 0 =
~


p Lu


vαβ  
vp


v 0 vα  
q Lr
−


vαβ  
vp

v 0 v β   − q Lq v 0
p Lu 
)
val (
−
+

vαβ   v p vαβ
v p 
+ 
 v11
 iC1 
 v1 
i  = p L v  − PLu v +  + 1 p L 0
Vectorial  C 2 
2
 21 
2
v 2   V1+ 2  + 
3 v0
iC 3 
v3 
v31
 
v0 
v 2 − v3 
v  + 1 q L  v − v 
3
1
2
 0
3 v 
v0 
v
v
−
 1 2 
v0 PLu
v02αβ
PC = 0
0
PC = 0
0
PC = 0
3. Simulation Results Evaluation.
To carry out a comparative analysis among the five compensation strategies got from the five
formulations presented, it has been designed a simulation platform to implement the five APLC
control algorithms, at three operation conditions. The power system is a three-phase four-wire load,
built by two bi-directional SCRs and a serial resistor in each phase. Different values have been
assigned to the resistors in each phase to make up an unbalanced load. Three different source voltages
have been applied to the load: balanced sinusoidal, unbalanced sinusoidal, balanced non-sinusoidal.
Figure 1 shows the load and source currents after compensation for the five control strategies in the
case of balanced and sinusoidal voltage. In this case the five control algorithms get sinusoidal and
balanced source current although the Park’s one is not in phase with the voltage. It must be considered
that id and iq dc components represent the positive-sequence first harmonic current and not the active
current component.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.4
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
7
6.
5
6
5.
5
5
4.
5
10
4
8
3.
5
3
6
2.
5
4
2
2
0
-2
-4
-6
-8
-10
p-q
d-q
Modified
p-q-r
Vectorial
Before comp.
Figure 1: Source current before and after compensation in Case 1: Sinusoidal and Balanced Source
Voltage.
Figure 2 presents the phase-one currents in the case 2: sinusoidal unbalanced voltages. There is a
considerable difference among the results got from such formulation. Only two of them, vectorial and
d-q formulations achieve to get a sinusoidal current waveform. Nevertheless, d-q formulation still
presents a phase difference respect to the voltage due to the reasons mentioned above.
7
6.
5
6
5.
5
5
10
4.
5
4
8
3.
5
6
3
4
2.
5
2
2
0
-2
-4
-6
-8
-10
p-q
d-q
Modified
p-q-r
Vectorial
Before comp.
Figure 2: Source current before and after compensation in the Case 2: Sinusoidal and Unbalanced
Source Voltage.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.5
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
Figure 3 presents the phase-one currents in the case 3: non-sinusoidal balanced voltages. In this case,
in a clearer way than in case 2, it can be observed that only vectorial and d-q formulations achieve to
get a sinusoidal current. Once more, d-q formulation presents a phase difference respect to the voltage.
7
6.
5
6
5.
5
5
10
4.
5
4
8
3.
5
6
3
4
2.
5
2
2
0
-2
-4
-6
-8
-10
p-q
d-q
Modified
p-q-r
Vectorial
Before comp.
Figure 3: Source current before and after compensation in Case 3: Non Sinusoidal and Balanced
Source Voltage.
Figure 4 represents, through a bar diagram the neutral current rms value in the three cases. All the
control strategies achieve to eliminate the neutral current except the modified p-q formulation.
9
8
7
RMS
6
5
4
3
2
1
0
p-q
d-q
Sinusoidal balanced
Modified
p-q-r
Sinusoidal unbalanced
Vectorial
Before
Comp.
Non sinusoidal balanced
Figure 4: Neutral Current RMS
Figures 5 and 6 present, through bar diagrams, a quantitative analysis of the results respect to the
distortion. Figure 5 shows the load and source currents after compensation THD (total harmonic
distortion) value defined in (3) and got from the five formulations in the three cases considered.
The five formulations present a null distortion in the case 1. In cases 2 and 3 only vectorial and d-q
formulations present a cero THD value. In case 2 p-q, modified p-q and p-q-r formulations present
similar results with a distortion about the 10%. In the case 3 p-q and p-q-r formulations present THDs
bellow the 10% and modified p-q over that value. Finally, in figure 6 it has been wanted to represent a
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.6
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
global distortion index which corresponds to the whole three phase system. It includes phase current
and neutral current, TDD3φ, defined as:
ITDD3φ = ITHD
ITHD1φ =
2
L1
I12φ − I12φ1
I12φ1
I L211
I e2
=
+ ITHD
I L221
2
L2
I e2
+ ITHD
2
L3
I L231
I e2
+ ITHD
2
L4
I L241
(2)
I e2
I12φ 2 + I12φ3 + I12φ 4 + ... + I12φ N
(3)
I12φ1
In case 1 all the formulations obtain a TDD3φ null, as could be wondered. In case 2, only the vectorial
and d-q formulations present a distortion value null, although d-q presents a phase difference between
source current and voltage as was mentioned above. P-q and p-q-r formulations present values not null
and in the same order, whereas modified one presents a bit higher value. In case 3, vectorial
formulation is again the only one which presents a null distortion value without phase difference
between source current and voltage. The p-q and p-q-r present again values in the same order whereas
the modified one presents a much higher value.
0,6
0,5
TDD1F
0,4
0,3
0,2
0,1
0,0
p-q
d-q
Modified
Sinusoidal balanced
p-q-r
Vectorial
Sinusoidal unbalanced
Before
Comp.
Non sinusoidal balanced
Figure 5: Phase 1 Current Distorsion Index.
0,8
0,7
ITDD3F
0,6
0,5
0,4
0,3
0,2
0,1
0,0
p-q
d-q
Sinusoidal balanced
Modified
p-q-r
Sinusoidal unbalanced
Vectorial
Before
Comp.
Non sinusoidal balanced
Figure 6: Three-Phase Quality Index.
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.7
APPLICATION OF THE INSTANTANEOUS POWER THEORIES IN LOAD COMPENSATION WITH ACTIVE POWER FILTERS
SALMERÓN PATRICIO
4. Discussion and conclusions.
After the comparative study carried out they could be fixed the next considerations:
• It is considered as compensation target the obtention of sinusoidal source current in phase with the
positive-sequence symmetrical component of the applied voltage fundamental harmonic, the
configuration used as ideal reference. At this conditions: P-q, modified p-q, p-q-r and vectorial
formulation 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 and modified p-q
does not get to clear the neutral current.
• Only vectorial and d-q formulations achieve to get a null distortion in all the cases. P-q and p-q-r
allow to obtain control algorithms in cases 2 and 3 with a distortion bellow the 10%. Modified p-q
goes over that value in case 3.
In summary, it can be said that only vectorial formulation is adequate to establish APLC compensation
strategies with any kind of load and any kind of supplies. Nevertheless, original formulation presents a
good performance, which can be improved, to look for adequate compensation strategies, if its
representation through the mapping matrix is changed by a vectorial representation.
References
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[2]. P. Salmerón, J.C. Montaño, Instantaneous Power Components in Polyphase Systems Under
Nonsinusoidal Conditions, IEE Proc.-Sci. Meas. Tech., Vol.143,No.2, March 1996.
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[8]. D.A. Marshall, F.P. Venter, J.D. van Wyk, An Evaluation of the Instantaneous Calculation of
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[9]. H. Kim, F. Blaabgerg, B. Bak-Jensen, J. Choi, Novel Instantaneous Power Compensation Theory
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[10]. H. Kim, F. Blaabjerg, B. Bak-Jensen and J. Choi, Instantaneous Power Compensation in ThreePhase Systems by Using p-q-r Theory, Conference Records of IEEE/PESC’01, June 2001.
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.8