A new approach for three-phase loads compensation basedon the instantaneous reactive power theory
Patricio Salmeron ∗, Reyes S. Herrera, Jesus R. VazquezElectrical Engineering Department of the University of Huelva, Spain
Received 6 July 2006; received in revised form 8 March 2007; accepted 9 May 2007Available online 27 June 2007
bstract
The original instantaneous reactive power theory or p–q theory has been systematically used in the control of active power filters (APFs). Whenhe APF is connected in parallel to a non-linear and unbalanced load, the p–q theory application has allowed a compensation strategy namedonstant power to be obtained. This means that, after the APF connection, the instantaneous power supplied by the source is constant and itas the same value as the average power consumed by the load. Nevertheless, the use of other compensation strategies is possible: unity poweractor or sinusoidal and balanced supply currents, among others. This paper shows that any compensation strategy may be developed into the
–q theory frame. Besides, the paper presents a p–q theory reformulation without using mapping matrices, which makes easier the obtention ofhe compensation currents. Finally, an exhaustive analysis of practical cases has been carried out at simulation and experimental level through aaboratory prototype which has allowed the proposed approach to be verified.
2007 Elsevier B.V. All rights reserved.
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eywords: Active power filters; Harmonics; Instantaneous reactive power; Pow
. Introduction
The integration of power electronics in the commercial andndustrial processes has originated a considerable increase ofon-linear currents. The problems associated with these dis-orted currents have made of harmonics compensation a mainask. There are different equipments to achieve the non-linearurrents compensation. One of the most used currently is thective power filter (APF) [1–5]. An APF is a power electron-cs converter which supplies the non-useful current required byhe load, making possible that the source only supplies the use-ul power required by the load. The active power filter controls based on the instantaneous reactive power theory or otherormulations later proposed [6–15].
Akagi et al. [6] introduced the instantaneous reactive powerr p–q theory in the control of the APFs. With this formulation,
ts authors got to design an APF control circuit in an easy way. Inhis theory approach, the more ambitious objective is achievedhen the instantaneous power supplied by the source is constant
nd with the same value as the average power consumed by theoad. This has been named constant power compensation. If theource voltage is balanced and sinusoidal, the constant powerompensation allows a balanced and sinusoidal source currento be obtained. If the source voltage is non-sinusoidal and/ornbalanced, the current will not be balanced and sinusoidal afterompensation. This point has been one of the aspects more crit-cized of the p–q theory. Nevertheless, certainly, the p–q theoryas been the most used till nowadays in the APFs control.
This paper demonstrates that, with a reformulation of the p–qheory, all the compensation objectives can be obtained: unityower factor compensation or balanced and sinusoidal sourceurrent strategy [7]. And this is true for any source voltageondition.
The p–q theory has been formulated from its origin referringhe voltage and current variables in the 0αβ reference sys-em obtained after applying the Clark transformation [8]. So,he compensation currents are obtained by means of succes-ive application of different mapping matrices. This makes truly
ard the mathematical analysis when other control strategies arepplied.
In this paper, a p–q theory reformulation has been carried out.t avoids the mapping matrices, and it makes use of the vectorial
mmecphave been always limited to the constant power strategy. Actu-ally, the vectorial frame is a conceptual approach. So, three newvoltage vector will be defined, �e0αβ : �e0, �eαβ and �e−βα, Fig. 2.
06 P. Salmeron et al. / Electric Power
epresentation, which simplifies the obtention of compensationurrents.
Finally, the different compensation strategies have beenpplied to a practical case, which has allowed the proposedpproach to be verified.
The paper is organized as follows. In Section 2, the p–qheory is summarized and a new formulation, without using
apping matrices, is presented. Section 3 presents the way ofbtaining the compensation currents in the p–q theory referencerame according to the strategy denominated constant power. Inection 4, the procedure to obtain the compensation currentsith the compensation objective named unity power factor is
stablished. In Section 5, a similar development with the com-ensation objective of obtaining balanced and sinusoidal sourceurrents is presented. The two last control strategies have noteen developed up now in the p–q original theory frame. Finally,ection 6 presents the simulation results corresponding to the
hree compensation strategies applied to a practical case and Sec-ion 7 shows the results obtained by applying the three strategieso an experimental laboratory prototype.
. Original p–q theory
The instantaneous reactive power theory was formulated athe beginning of the eighties [6], and this is the formulationith the largest diffusion along these years. For that reason, it
s the most used as control strategy in the APF. The p–q theoryas developed for three-phase three-wire systems with balanced
nd sinusoidal source voltages. The compensation objectivessumed was the obtention of a constant instantaneous sourceower.
The theory is based on a translation from the phase referenceystem (123) to the 0αβ system, Fig. 1.
The transformation matrix associated is as follows:
e0
eα
eβ
⎤⎥⎦ =
√2
3
⎡⎢⎢⎢⎢⎢⎢⎣
1√2
1√2
1√2
1 −1
2−1
2√ √
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎣
u1
u2
u3
⎤⎥⎦ (1)
03
2− 3
2
Fig. 1. 0αβ referente system.
ms Research 78 (2008) 605–617
i0
iα
iβ
⎤⎥⎦ =
√2
3
⎡⎢⎢⎢⎢⎢⎢⎣
1√2
1√2
1√2
1 −1
2−1
2
0
√3
2−
√3
2
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎣
i1
i2
i3
⎤⎥⎦ (2)
rom Eqs. (1) and (2), it can be deduced that
N = i1 + i2 + i3 = i0√
3 (3)
The different power terms are defined as follows:
p0
pαβ
qαβ
⎤⎥⎦ =
⎡⎢⎣
e0 0 0
0 eα eβ
0 −eβ eα
⎤⎥⎦⎡⎢⎣
i0
iα
iβ
⎤⎥⎦ = [T ]
⎡⎢⎣
i0
iα
iβ
⎤⎥⎦ (4)
here p0 is the zero sequence (real) instantaneous power, pαβ
he αβ real instantaneous power and qαβ is the imaginary instan-aneous power.
Considering the [T] inverse matrix, the calculation of the cur-ent components from the different power terms is possible. Thexpression is shown in the next equation:
i0
iα
iβ
⎤⎥⎦ = 1
e0e2αβ
⎡⎢⎣
e2αβ 0 0
0 e0eα −e0eβ
0 e0eβ e0eα
⎤⎥⎦⎡⎢⎣
p0
pαβ
qαβ
⎤⎥⎦ (5)
here e2αβ = e2
α + e2β
Although original p–q theory is developed from the mappingatrices (4) and (5), an alternative develop is possible. This newodel does not use any mapping matrices. Therefore, it makes
asier the theory treatment and its application to the active filtersontrol. In fact, the vectorial frame allows to obtain, in a sim-le way, any compensation strategy, while the mapping matrices
Fig. 2. New voltage vectors in 0αβ coordinates system.
Systems Research 78 (2008) 605–617 607
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P. Salmeron et al. / Electric Power
The voltage and current space vectors are defined as follows:
αβ =
⎡⎢⎣
0
eα
eβ
⎤⎥⎦ ; �e−βα =
⎡⎢⎣
0
−eβ
eα
⎤⎥⎦ ; �v = �e0 =
⎡⎢⎣
e0
0
0
⎤⎥⎦ ;
�i =
⎡⎢⎣
i0
iα
iβ
⎤⎥⎦ (6)
here �e−βα is the orthogonal voltage vector and �e0 is the zeroequence voltage vector. It is always verified that
0αβ = [ e0 eα eβ ]t = �eαβ + �e0 (7)
nd
−βα · �e0αβ = 0 (8)
n the 0αβ reference frame, the current vector�i may be expresseds the sum of its projections over the vectors �e0, �eαβ and �e−βα,hat is
= pαβ(t)
�eαβ · �eαβ
�eαβ + qαβ(t)
�e−βα · �e−βα
�e−βα + p0(t)
�e0 · �e0�e0
= pαβ(t)
�eαβ · �eαβ
�eαβ + qαβ(t)
�eαβ · �eαβ
�e−βα + p0(t)
�e0 · �e0�e0 (9)
here pαβ = �eαβ · �i is the instantaneous real power in α–β com-onents, qαβ = �e−βα · �i the instantaneous imaginary power and0(t) is the zero sequence instantaneous real power. They aredentical to those power terms defined in Eq. (4). The fact that therthogonal voltage vector norm and the voltage vector withoutero-sequence component norm are the same has been consid-red in Eq. (9). Since now the compensation currents will bebtained from several strategies within the model presented inhe Eq. (9), without using mapping matrices.
. Constant power compensation
The strategy assumed by the original or p–q theory since itsrigin has been the obtention of a constant instantaneous powern the source side with the only restriction of getting a nullverage instantaneous power exchanged by the compensator,c. Along the paper, lower case represents instantaneous val-es, upper case average values, subindex “L” load requirement,ubindex “C” compensator supply and subindex “S” source sup-ly.
In fact, total power required by the load can be expressed asollows:
here uppercase are referred to the average values and the termsith the character ∼ over it are referred to the power oscillatory
omponent.
To calculate the compensator current, and according to Fig. 3,
t is verified that
C(t) = pL(t) − pS(t) = pL(t) − PL (11)
s
p
Fig. 3. Power system compensated by an active power line conditioner.
here PL is the total active power incoming to the load.S(t) = PL after compensation. Taking into account Eq. (10),nd the independence of 0αβ coordinates, the Eq. (11) can bexpressed in the next way:
Cαβ(t) = pLαβ(t) − PLαβ = pLαβ(t) (12)
C0(t) = pL0(t) − PL0 = pL0(t) (13)
hese equations, effectively, means that the average valuepC(t)〉 = PC = 0, where pL(t) represents the ac or oscillatory partf pL(t).
On the other hand, the instantaneous imaginary powerxchanged by the compensator must be the same as the instan-aneous imaginary power required by the load:
C(t) = qLαβ(t) (14)
Therefore, from Eq. (5), the compensation current is
iC0
iCα
iCβ
⎤⎥⎦ = 1
e0e2αβ
⎡⎢⎣
e2αβ 0 0
0 e0eα −e0eβ
0 e0eβ e0eα
⎤⎥⎦⎡⎢⎣
pL0(t)
pLαβ(t)
qLαβ(t)
⎤⎥⎦ (15)
his strategy achieves a constant instantaneous power suppliedy the source. Nevertheless, it does not eliminate the neutralurrent.
After compensation, the source supplies a constant instanta-eous power with a value identical to the load active power withneutral current not null.
The results presented in Eq. (15) have been got by mean ofhe mapping matrices introduced by Akagi et al. [6].
In the new model of the p–q theory, without the use of map-ing matrices, the constant power compensation strategy cane developed. Besides, the elimination of the neutral current isossible [9]. The corresponding procedure is presented now.
After compensation, the source current �iS(t) is the next:
S(t) = K�eαβ (16)
Taking into account that �e0 · �eαβ = 0, from Eq. (6) to (7), theource total instantaneous real power must be as follows:
mposing that pS(t) = PL, the proportionality factor can bebtained and it gets the next value:
= PL
e2αβ
(18)
herefore, from Eqs. (9) and (18), the compensation currentsay be expressed like a vector as follows:
C(t) = �iL(t) − iS(t)
= pαβ(t)
e2αβ
�eαβ − PL
e2αβ
�eαβ + pL0(t)
e20
�e0 + qLαβ
e2αβ
�e−βα
= pαβ(t) − PL0
e2αβ
�eαβ + pL0(t)
e20
�e0 + qLαβ
e2αβ
�e−βα (19)
here pαβ(t) represents the oscillatory part of the instantaneouseal power in the α–β plane.
From Eq. (19), such component of the source current is asollows:
c0 = pL0(t)
e20
e0 (20)
cα = pLαβ(t) − PL0
e2αβ
eα − qLαβ
e2αβ
eβ,
cβ = pLαβ(t) − PL0
e2αβ
eβ + qLαβ
e2αβ
eα (21)
. Unity power factor compensation
The unity power factor compensation is a strategy widelysed along the time. The target is to obtain source currents withhe same distortion and symmetry condition as source voltages.t means that the source current is collinear to the supply voltage.n this situation, the source supplies the load active power, buthe instantaneous real power is not constant after compensation7].
When the source voltages are sinusoidal and balanced, con-tant power and unity power factor compensations results arehe same. In fact, for sinusoidal and balanced source voltages,f compensation currents cancel reactive, distortion and unbal-nced current components, source currents will be sinusoidal andalanced in the same way as voltage. And for that reason, instan-aneous power supplied by the source will be constant. However,hen the supply voltage is unbalanced and non-sinusoidal the
ituation won’t be the same.Since now, a control strategy will be obtained. It is based on
he p–q theory. This new strategy allows unity power factor inhe source side to be obtained after compensation:
s = Ge�e0αβ (22)
Taking into account Eqs. (11) and (22) and to satisfy that thective power exchanged by the compensator is null:
C(t) = pL(t) − pS(t) = pL(t) − Gee2(t) (23)
i
v
ms Research 78 (2008) 605–617
hus, knowing that 〈pC(t)〉 = PC = 0 the proportionality constants as follows:
e = PL
E2 (24)
ince 〈pL(t)〉 = PL and where E2 is the square root of voltageector RMS value:
2 = 1
T
∫T
e20αβ dt (25)
inally, after compensation:
iS0
iSα
iSβ
⎤⎥⎦ = PL
E2
⎡⎢⎣
e0
eα
eβ
⎤⎥⎦ (26)
From Eq. (11) and considering the reference axes, compen-ation power can be expressed as follows:
C0(t) = pL0(t) − pS0(t) = pL0(t) − PL
E2 e20 (27)
Cαβ(t) = pLαβ(t) − pSαβ(t) = pLαβ(t) − PL
E2 e2αβ (28)
Besides, instantaneous imaginary power verifies:
C(t) = qLαβ(t) (29)
o,
C = �iL −�iS = pLαβ(t)
e2αβ
�eαβ − PLαβ
E2αβ
�eαβ + p0(t)
e20
�e0 − PL
E20
�e0
+ qLαβ(t)
e2αβ
�e−βα (30)
In this way, the explicit expressions of the compensationurrents in the 0–α–β coordinates are as follows:
iC0 =(
p0(t)
e20
− PL
E20
)e0,
Cα =(
pLαβ(t)
e2αβ
− PLαβ
E2αβ
)eα − qLαβ(t)
e2αβ
eβ,
Cβ =(
pLαβ(t)
e2αβ
− PLαβ
E2αβ
)eβ + qLαβ(t)
e2αβ
eα (31)
This control algorithm does not permit to eliminate the neutralurrent, since:
p0(t) p0(t) PL PL
S0 = iL0 − iC0 =
e20
e0 −e2
0
e0 +E2
0
e0 =E2
0
e0 (32)
This expression will be different of zero if zero-sequenceoltage component exists.
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. Sinusoidal and balanced source currentompensation
For last years, both compensation objectives presented in thisaper have not been enough to solve the new problems associatedo the voltage and current distortion. It is necessary to define aew compensation objective. It is the sinusoidal source currentompensation strategy. The aim is to obtain a sinusoidal andalanced source current, in phase with voltage [11–13], whenhe voltage is sinusoidal and balanced or in phase with voltageositive sequence fundamental component in any other case. Itust be fulfilled in any supply voltage conditions and load kind.If voltages are balanced and sinusoidal, currents proposed
ill be proportional to the voltages (as in previous strategy).he proportionality constant must have the value established in
he Eqs. (24) and (25), too.If voltages are balanced non-sinusoidal, the reference source
urrent must be proportional to its fundamental component. Theroportionality constant is fixed to satisfy the restriction of hav-ng null active power exchanged by the compensator. The idealurrent expression is as follows [12,13]:
iS0
iSα
iSβ
⎤⎥⎦ = PL
E21
⎡⎢⎣
0
eα1
eβ1
⎤⎥⎦ (33)
α1 and eβ1 represent the fundamental components of the α, β
oltage, respectively, and E21 represents the square root of the
oltage �eαβ1 fundamental component RMS value.To analyze the case of unbalanced non-sinusoidal voltage, it
s necessary to consider that the vector �eαβ from Fig. 2 may beivided in the positive sequence fundamental component �e+
αβ1,
egative sequence fundamental component �e−αβ1, and harmonic
omponents �eαβN1. Current vector can be expressed, in this way,s follows:
L = pLαβ(t)
e2αβ
�eαβ + PL0
e20
�e0 + qLαβ(t)
e2αβ
�e−βα
= pLαβ(t)
e2αβ
�e+αβ1 + pLαβ(t)
e2αβ
�e−αβ1 + PLαβ(t)
e2αβ
∑∀n≥2
�eαβn
+PL0
e20
�e0 + qLαβ(t)
e2αβ
�e−βα (34)
So, in the case of unbalanced non-sinusoidal voltage, theource current expression becomes as shown in the followingquation:
iS0
iSα
iSβ
⎤⎥⎦ = PL
E+21
⎡⎢⎣
0
e+α1
e+β1
⎤⎥⎦ (35)
+21 represents the square root of the voltage positive sequence
undamental component RMS value.In (35), iSi is proportional to the voltage positive sequence
undamental component. The proportionality constant value
poal
ms Research 78 (2008) 605–617 609
s calculated imposing the restriction of null active powerxchanged by the compensator.
Thus, the compensation current equations to obtain balancednd sinusoidal supply currents are calculated, from Eq. (35), asollows:
C = �iL −�iS =(
pLαβ(t)
e2αβ
− PL
E+2αβ1
)�e+αβ1 + pLαβ(t)
e2αβ
�e−αβ1
+PLαβ(t)
e2αβ
∑∀n≥2
�eαβn + PL0
e20
�e0 + qLαβ(t)
e2αβ
�e−βα (36)
In this case, it is necessary to impose restrictions not onlyo the instantaneous real powers p0(t) and pαβ(t), but besides tohe instantaneous imaginary power qαβ(t). This new restrictions necessary to get sinusoidal and balanced source currents inhe case of non-sinusoidal or unbalanced voltages.
In fact:
Cαβ = �e−βα · �iC = − PL
E+2αβ1
�e−βα · e+αβ1 + qLαβ(t)
e2αβ
e2αβ
= − PL
E+2αβ1
�e−βα · e+αβ1 + qLαβ(t) (37)
ince in general �e−βα · e+αβ1 = 0
In mapping matrices format, the control strategy may bexpressed as
iC0
iCα
iCβ
⎤⎥⎦ = 1
e0e2αβ
⎡⎢⎣
e2αβ 0 0
0 e0eα −e0eβ
0 e0eβ e0eα
⎤⎥⎦
×
⎡⎢⎢⎢⎢⎢⎣
pL0(t)
pLαβ(t) − PL
E+21
(eαe+α1 + eβe+
β1)
qLαβ(t) − PL
E+21
(eαe+β1 − eβe+
α1)
⎤⎥⎥⎥⎥⎥⎦ (38)
Moreover, this strategy achieves to eliminate the neutral cur-ent with a null active power exchanged by the compensator.
. Simulation results
In this paper, the different control strategies proposed withinnstantaneous reactive power theory have been applied to a three-hase four-wire system similar to the presented in Fig. 3. Its a three-phase four-wire system composed of an unbalancednd non-sinusoidal voltage source that supplies a non-linearnbalanced load. The compensator injects the non-useful currentequired by the load.
The three strategies presented in previous sections have beenpplied to the simulation platform shown in Fig. 4. This figure
resents a Matlab-Simulink implementation diagram. On thene hand, the source is composed of three single-phase sourcesnd a serial resistor in star configuration. On the other hand, theoad is composed of two back to back SCRs and a serial resistor,
610 P. Salmeron et al. / Electric Power Systems Research 78 (2008) 605–617
k implementation diagram.
wasphvTtcwrFc
ptF
b
e
f
vf
atavT
Fig. 4. Matlab-Simulin
hose value is different in each phase, in star connection. Thus,non-linear load is obtained. The main objective is the control
trategies analysis to calculate the reference current. So, for theurposes of the simulation analysis, a relatively simple modelas been considered for the compensator. Therefore, the con-erter is modeled by means of three controlled current sources.he compensation current is obtained through a set of calcula-
ion blocks, and they are directly injected in the system by theompensator. The block identified as “control block” is a maskhose content implements the different specified control algo-
ithm. All the algorithms used in this paper are summarized inig. 5, where uR is the voltage used to calculate the referenceurrent by each compensation strategy.
The method used to calculate the voltage fundamental com-onent, necessary in the last implemented strategy, is based onhe use of low band filters, multipliers and adders, as shown inig. 6.
In fact, any voltage or current signal, in αβ coordinates, maye expressed in the next way:
The product of the voltage component eα(t) and the wave-orm sin ωt has a constant term: 1/2Eα1 cos ϕ1 being Eα1 the
fwtf
Fig. 5. Control block alg
Fig. 6. Method to get voltage fundamental component.
oltage fundamental component amplitude and ϕ1 is the voltageundamental component phase.
On the other hand, the product composed by the same volt-ge component eα(t) and the waveform cos ωt has a constanterm: 1/2Eα1 sin ϕ1. The product of both constant terms per 2nd per sin ωt and cos ωt, respectively, compose the phase 1oltage fundamental component waveform, as shown in Fig. 6.he angular speed used to generate the sine and the cosine
unctions corresponds to the nominal pulsation of the voltage
aveform fundamental component. It is because the objective is
o extract the voltage fundamental component waveform, whoserequency is the nominal one.
orithms diagram.
P. Salmeron et al. / Electric Power Syste
pwtvvv
bafvc
srcsai
6
rps
6
aa
aaaascsttcrtuttspb
a
6
ta
tppssrAsab
bp
Fig. 7. Supply voltages (V), used in simulation assays.
The results got from the different simulation strategies areresented in Figs. 7–9 which show two periods (0.04 s) of theaveforms. Fig. 7 presents the voltages supply correspondent
o each case. The first graph shows a balanced and sinusoidaloltage system, the second one an unbalanced and sinusoidaloltage system and the last one a balanced and non-sinusoidaloltage system.
The results obtained from each strategy application in case 1,alanced and sinusoidal supply voltage, are identical: balancednd sinusoidal source current. Fig. 8 presents the current wave-orms which are obtained applying an unbalanced and sinusoidaloltage system, i.e., the case 2. And, finally, Fig. 9 presents thease 3, balanced and non-sinusoidal supply voltage.
The first graph in Figs. 8 and 9 present the three-phaseource current before compensation, i.e., the three-phase currentequired by the load. It is a strongly non-linear and unbalancedurrent even if the voltage supply system is balanced and sinu-
oidal. Of course, the load currents present a different distortionnd unbalanced behavior for each supply condition correspond-ng to each case.
cato
ms Research 78 (2008) 605–617 611
.1. Case 1: balanced and sinusoidal voltage supply
When the supply voltage is balanced and sinusoidal, theesults obtained applying the three control strategy (constantower, unity power factor and sinusoidal source current) are theame.
.2. Case 2: unbalanced and sinusoidal voltage supply
In this case, load current shown in Fig. 8 first graph is unbal-nced due to the load unbalance effect and to the supply voltagesymmetry, simultaneously.
The second graph presents the three-phase source currentfter compensation when the strategy named constant power ispplied. This graph shows a source current distortion consider-bly reduced. This strategy does not get sinusoidal source currentlthough it obtains constant instantaneous power supplied by theource after compensation. The third graph presents the sourceurrent after compensation according to the unity power factortrategy. In this case, the source current waveform is collinear tohe voltage and the strategy is not able to compensate the part ofhe load current due to the supply voltage zero sequence-phaseomponent. So, this strategy does not eliminate the neutral cur-ent and it does not get constant instantaneous power supplied byhe source after compensation. However, the power factor is thenity. The three-phase current waveform shown in last graph ishe only one which is balanced and sinusoidal. It corresponds tohe sinusoidal source current strategy that imposes getting sinu-oidal and balanced source currents in phase with the voltageositive sequence phase and carrying the active power requiredy the load.
However, the power supplied by the source is not constantnd the power factor is not exactly the unity.
.3. Case 3: balanced and non-sinusoidal voltage supply
In this case, the load current shown in Fig. 9 is strongly dis-orted due to the simultaneous action of the load nor linearitynd the applied voltage distortion.
As in the two earlier cases, in Fig. 9 second to fourth graphshe three-phase source currents correspond to the different com-ensation strategies are presented. The constant power oneresents a low distortion although the waveforms are not sinu-oidal. The third graph presents the results got applying thetrategy named unity power factor, where the source currenteproduces the voltages distortion. The power factor is the unity.t last, the fourth graph presents the results obtained using the
inusoidal source current strategy. The source current is now bal-nced and sinusoidal. The load unbalanced and distortion haveeen compensated.
In summary, a strongly non-linear and unbalanced load haseen considered to be studied in three different cases of sup-ly voltage. In case 1, the load unbalanced effect is completely
ompensated by all the control strategies. Nevertheless, if volt-ge is unbalanced and/or non-sinusoidal, each strategy achievehe target imposed in its design although they do not achieve anyf the other two, e.g., constant power strategy obtain constant
612 P. Salmeron et al. / Electric Power Systems Research 78 (2008) 605–617
Fa
pao
7
monb
Fa
(facegd
ig. 8. Three-phase source current (A) before/after compensation with unbal-nced and sinusoidal source voltages obtained in simulation assays.
ower supplied by the source, the power factor is not the unitynd the source current is not balanced and sinusoidal. And son, applying the other two compensation strategies.
. Experimental results
In this work, the APF control has been implemented by
eans of a specific digital signal processor (DSP) board devel-
ped by dSPACE. It includes a real-time processor and theecessary In/Out interfaces that allow the control operation toe carried out. In particular, DS1103 peer to peer connection
acs
ig. 9. Three-phase source current (A) before/after compensation with balancednd non-sinusoidal source voltages obtained in simulation assays.
PPC) controller board is equipped with a Power PC processoror fast floating point calculation at 400 MHz. This hardwarellows to program via Simulink. In this way, all the control cir-uit components are configured graphically within the Simulinknvironment. The RTI translates the Simulink model to C lan-uage, it generates the real-time executable program, and itownloads it in the controller board.
To check the approaches proposed in this paper, they werepplied to the three-phase four-wire unbalanced ac-regulatorompensation. Fig. 10 shows a general scheme of compensatedystem.
P. Salmeron et al. / Electric Power Systems Research 78 (2008) 605–617 613
e of the compensated system.
TtspSitiaaccAs2
tppPsubc
ra
sThe voltage applied is the net voltage unbalanced using an
autotransformer. The three phases root mean square (RMS) val-ues are indicated in Table 1 first row. These values point out agreat level of unbalanced applied to the system.
Table 1System parameters
Phase 1 Phase 2 Phase 3
Fig. 10. A general schem
The voltage source is constituted by three autotransformer.hey allow the voltage to be unbalanced before applying it to
he system. The load is composed of three regulators with aerial inductive load in each phase connected in star. The com-ensator is constituted by a Semikron power inverter modelKM50GB123, with a three-phase IGBT bridge and two capac-
tors of 2200 �F in dc side. In each phase, the connectionransformers voltage ratio is 230/460 V and the inverter outputnductance is Ls = 17 mH. The use of the coupling transformersllows working with a low dc voltage. The control block inputsre nine: the load voltages and currents, and the compensationurrents. Besides, it is necessary another input to control theapacitor dc voltage. Additional inputs are used to check thePF compensation performance. The voltage and current sen-
ors used are AD/DC LEM LA-35 NP and AC/DC LEM LV5-600 models, respectively.
The software running in the real-time processor carries outhe control. It calculates the reference source current as it wasresented in Sections 2–4. The difference between the real com-ensation current and the calculated before is the input to theWM module. Its output are the power circuit IGBTs trigger
ignals. In fact, in an experimental develop the current controlsed is very important. In our experimental system, a hysteresisand PWM control has been implemented. The converter IGBTsommutation introduces unavoidable high order harmonics. To
RRIS
Fig. 11. Control system used in the experimental set-up.
educe its influence, the PWM control is carried out by mean ofspecific circuit instead of by software, Fig. 11.
The RMS voltage values and the load parameters values arehown in Table 1.
MS voltage 203.84 147.81 221.92esistor (�) 52.2 52.2 52.2
nductance (mH) 150 150 150CRs angle 90 90 90
614 P. Salmeron et al. / Electric Power Systems Research 78 (2008) 605–617
Fig. 12. Phase 1 voltage (V) and current (A) obtained in the laboratory proto-type. (a) Before compensation, (b) after compensation applying constant powerstrategy, (c) after compensation applying unity power factor strategy and (d)after compensation applying sinusoidal source current compensation.
Fig. 13. Three-phase waveforms obtained in the laboratory prototype. (a) Volt-age applied, (b) source current before compensation, (c) source current aftercompensation applying constant power strategy, (d) source current after com-pensation applying unity power factor strategy and (e) source current aftercompensation applying sinusoidal source current compensation.
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P. Salmeron et al. / Electric Power
Besides, the feedback method used to control self-supportingc bus is very important. It supposes, in general, a modification ofompensation currents. Summarizing, the procedure used con-ists of controlling the capacitor voltage to a reference value.o achieve it, a PI control may be chosen for the error between
he reference value and the capacitor voltage value at the endf each period. The result is the value of the compensator lostower. This value modifies the power terms which appear in theompensation currents expressions corresponding to each strat-gy. The modification is in the next way: referring to the firsttrategy, Eqs. (20) and (21), the power lost is subtracted to the
˜ Lαβ term in the expressions of icα and icβ; referring to the sec-nd strategy, Eq. (31), the power lost is added to the PLαβ termn the expressions of icα and icβ; referring to the third strategy,q. (38), the power lost is added to the load power PL.
Fig. 12 presents the phase one source voltage and currentefore and after compensation. Fig. 12(a) graph shows the sit-ation before compensation, i.e., phase one voltage and loadurrent. Fig. 12 graphs (b), (c) and (d) show the results got by allhe three control strategies: constant power, unity power factornd sinusoidal source current, respectively. The phase differ-nce between voltage and current is eliminated by applying thehree strategies. In fact, the first graph shows a phase differenceetween both waveforms. It is not in the other three graphs.
Fig. 13 presents the three-phase waveforms correspondingo: voltage applied (graph 13 (a)), load current or source currentefore compensation (graph 13 (b)), and source current afterompensation using each control strategy (graphs 13 (c–e)): con-tant power, unity power factor and sinusoidal source current,espectively. Fig. 13(a) shows the voltage unbalanced. Fig. 13(b)resents an unbalanced and distorted three-phase load currentue to the load non-linearity and to the source voltage unbalance.
The source current distortion after compensation using anyf the three control strategy is much lower than the presentedy the load current. The unity power factor strategy does notet balanced source current and it does not eliminate the neutralurrent.
The sinusoidal source current strategy obtains balanced andinusoidal source current after compensation, Fig. 13(e). Eveno, a ripple in the waveforms is unavoidable due to the thresholdand imposed by the PWM control. The source current corre-
ponding to the constant power strategy presents a considerablyistortion and the corresponding to the unity power factor strat-gy shows the same unbalance as the voltage applied to theystem.
able 2xperimental results, voltage values (V) and currents values (A)
RMS
Phase 1 Phase 2 Phase 3 Neutra
oltage 203.84 147.81 221.92 69.04
ource currentBefore compensation 1.25 0.91 1.43 1.02Constant power 1.04 1.00 1.07 0.19Unity power factor 1.08 0.79 1.17 0.39Sinusoidal source current 1.06 1.03 1.08 0.19
as
s
ms Research 78 (2008) 605–617 615
The voltage RMS and total distortion demand (TDD) valuesnd the source current RMS and TDD values before and afterompensation are presented in Table 2. The TDD is calculateds follows:
DD =√
h22 + h2
3 + · · · + h2n
RMS2 (40)
here hi represent the order i harmonic RMS value and RMSepresents the waveform RMS value.
Besides, Table 2 presents the three-phase RMS value corre-ponding to voltage and currents. It is defined as follows [16]:
e =√
V 210 + V 2
20 + V 230
3, Ie =
√I2
1 + I22 + I2
3 + I24
3(41)
here Vi0 represents the phase voltage corresponding to thehase i and Ii represents the phase i current RMS value. Phasecorresponds to the neutral current.This table shows as the three control strategies decrease the
urrent distortion considerably. Except the values correspondingo the phase two, the distortion values are very similar applyinghe three control strategies. However, that distortion representsigh order harmonics in the case of sinusoidal current com-ensation and low and high order harmonics in the other two,ig. 13. It is corroborated by the values corresponding to thehase two. The values achieved applying sinusoidal current isower than the achieved in the other two phases. In an ideal case,hey would be null, but in this case the PWM block has a greatnfluence on the results, and therefore the compensation currentoes not track its reference exactly. On the other hand, the sourceurrent after compensation applying the control strategy derivedrom constant power compensation presents a distortion appre-iatively bigger than the corresponding to the sinusoidal sourceurrent strategy as can be corroborated by Fig. 13. The unityower factor compensation, does not obtain balanced sourceurrents.
Respect to the three-phase RMS values, they are very sim-lar in the case of applying constant power and unity poweractor compensation strategies and a little bigger in the casef sinusoidal source current. It is necessary to obtain balanced
nd sinusoidal source current if voltage is not balanced andinusoidal.
The column “Power factor” in Table 2 shows that the threetrategies improve the power factor value after compensation.
616 P. Salmeron et al. / Electric Power Systems Research 78 (2008) 605–617
Table 3Source voltage and current harmonic content (% over the fundamental component) before and after compensation
Bt
t
8
itrtc(roihrasit
baoge
R
esides, the corresponding to the unity power factor strategy ishe highest of them.
Table 3 presents the main harmonic order corresponding tohe source current three phases before and after compensation.
. Conclusions
This paper has developed an exhaustive analysis of the p–qnstantaneous reactive power theory based on a new formula-ion without using mapping matrices. The calculation of theeference compensation currents is based on the current vec-or projections over three new voltage vectors defined in 0αβ
oordinates system. The three main compensation objectivesconstant power, unity power factor and sinusoidal source cur-ent) have been studied to generate three control strategies basedn the p–q instantaneous reactive power theory, one correspond-ng to each compensation objective. These control strategiesave been implemented in a simulation platform to study theesults obtained in three cases of voltage applied: sinusoidal
nd balanced, sinusoidal but unbalanced and balanced but non-inusoidal. Thus, the analysis shows that, with few modificationsn the control strategy developed by the authors of the originalheory, any compensation objective imposed to the system may
e achieved. So, the original p–q theory can be used as the base ofctive power filters control algorithm, to get any compensationbjective and with any voltage condition in the PCC. The strate-ies have been assayed by simulations in the Matlab-Simulinknvironment and tested in an experimental laboratory prototype.
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