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Energy Control of Three-phase Four-wire
Shunt Active Power Filter
P. Rodriguez, Student Member, IEEE, R. Pindado, Member, IEEE, and J . Pou, Member, IEEE
Absfrocl This p aper presents a three-phase four-wire active
power fiter controlled under an energy approach. For active
power Pter implementation, a nonconventional convertertopology is presented and analyzed. With this topology, and
considering harmonics and imbalances in utility voltage and load
current, power requirements on the active power filter arestudied. From this study, a controller based on the energy state of
the system is designed. In this paper, an analytical study and
verification by simulation me conducted.
Index Terms-Power condition ing, Pow er fit er s.
I. INTRODUCTION
N three-phase four-wire shunt active power filter
source inverter are commonly used, namely the four-leg full-
bridge (FLFB) topology, and the three-leg split-capacitor
(TLSC) topology. These topologies were presented at the
beginning of the 90 s 111, and numerous publications on their
control have appeared ever since [2]-[4]. The FLFB converter
shows better controllability thanks to its greater number of
semiconductor devices. However, interaction between the legs
connected to the utility phases and the leg connected to the
neutral conductor makes necessary space-vector based currentcontrol in order to achieve suitable reference current tracking.
The TLSC converter, having a smaller number of switches,
permits each of the three legs to be controlled independently.
making its current tracking control simpler than the previous
topology. H owever, in this case t he' zero-sequence injected
current flows through the dc-bus capacito rs. This current gives
rise to imbalance in the capacitors voltage sharing, forcing to
increase the capacitors rating to constantly ensure that the
capacitor voltages have a sufficiently high absolute value.
Merging FLFB and TLSC topologies, an altemative topology,
shown in Fig.1, can he obtained [ 5 ] . This four-leg split-
capacitor (FLSC) topology solves the cited problems of the
previous topologies.
I pplications, two topologies for current-controlled voltage
Manuscript received August 8, 2003. This work was supported by the
Comissionat lnterdepartamental de Recerca i lnnovaeib Tecnolhgica (CIRIT)
under theGrant DPl2001-2192.
P. Rodriguez is with the Electrical Engineering Depanment, Technical
University ofCatalonia, Barcelona, S P A N (e-mail: prodriguer@ ee.upc.es).
R. Pindado, and J. Pou are with the Electronics Engineering Department,
Technical University. of Catalonia, Barcelona, SPAIN (e-mail: pindado@
eel. pc.es, pa @eel.upc.es).
L I
Fig. 1. Active power filter based in FLSC converter
11. THREE-PHASEOUR-WIRE ACTNE FILTER AVERAGE MODEL
By means of a simple analysis of one leg in the circuit
shown in Fig. 1, and supposing Cl=C2=C generic switching-
leg can be represented by a state-space average model, in
which the state equation is (l a) and the output equation is (Ib).
In these equations d j E[O , ] is the duty cycle of the leg,
where i=jo,b,c,d). To simplify notation, the variables in theseequations represent averaged values over a switching period,
that is: iF,-c i , s, s ( , tc.
L A
Based on (Ib), the duty cycle di can be expressed as a
function of the leg voltages, obtaining d, = ( v n - c 2 ) vdc,
where vdc= vC,- Cz represents the de-bus absolute vol tage.
Substituting d , expression in current terms of I b), and taking
into account that the averaged value of leg output voltage is
vF i= vs ,+ L,i,, then the currents in capacitors arc given by
(2).
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equal zero. Besides, the converter cannot always meet the
(2a) dynamic requirements imposed by the reference phase
currents. In this situation, im will not be zero and instantaneous
imbalance appears in the dc-bus voltage sharing. For this
reason, an appropriate controller, presented in [ 6 ] , 7], should
be used so that, from sensing voltage imbalance in the
capacitors, the reference current for phase d an be modified inorder t o reach zero imbalanct:.
I
Vd'
1
vd
i,, = - ( v c 2 i , - v s , i F ,- L,,;~,;,,)
lcz = - ( - V c ; + V a i , +LF, iF jF< (2b)
Once the average model of the switching-leg is justified, theaverage model of the active filter is readily obtained by
connecting four averaged switching-legs and the rest of the
circuit components. From I ) , the four-wire model can be
depicted by the state equations shown in (3).
I v . ENERGY STORED NTHE DC-BUS
Taking into account (5 ) , the dc-bus energy variation,
Expression fo r current flowing through capacitors in Fig. 1
will be obtained by summing the terms shown in (2) for the
different legs, keeping in mind that v g 0 . These expressions
are shown in (5).where im is the injected current in the dc-bus
midpoint, and p F 3 / s the instantaneous active power develop ed
by the filter.
This expression evidences that the energy stored in the dc-
bus does not depend on the current injected in the dc-bus
midpoint, but rather, it only depends on the instantaneousactive power developed by the filter, pF3+ and the
instantaneous pow er associated to the coupling inductances of
the legs, pL F. In a real implementation, additional power
consump tions will appear in different components of the active
power filter mainly due to conduction and switching losses.
Considering these additional power losses, pims, 7 ) should be
modified to obtain that
Expression (8) is extremely simple, and it shows a linear
relationship between energy variation in the dc-bus and power
terms associated to the filter. For this reason, and assuming
that the fourth leg is operative in the converter, it seem s logical
easily made from the estimation of the energy stored in dc-bus.
Therefore, t?om an energy approach, the active power filter
i=o,b.c,d
i=o,b,c,d
- vc , i Fo+ p F 3 ( L, x iFjiF,)]5b) to think that the control of ihe dc-bus abs olute voltage will b e .
could be represented by means of the block diagram shown in
Fig. 2, where p , n t = p i m 5 p J y . n a later section, the controller
dedicated to eliminate variations in the &-bus will be
studied.
111. DIFFERENTRLOLTAGE IN THE DC-BUS
v C f ( 0 ) = -yCZ(o) and taking into account (51, the
dc-bus dferen f i a l vo l tage , Av c , will be given by (6).
1
cvc =- ; . ,+ c l ) dr
Equation (6) evidences, as logic, that the dc-bus differential
voltage lineally depend s on the injected current in the dc-bus
midpoint. Therefore, in the circuit shown in Fig. I , if
iFd= -( iFa+ Fb F c ) hen the dc-bus voltage imbalance will
be eliminated. At this point, it is worth noting that, although
the condition iLd =-( * + f F b* + i F c ) IS imposed for the
reference currents, this does not imply that this equality is
always true for the instantaneous currents, since the
instantaneous sum of the current ripples of the four legs is not
, I_ _ r
Fig. 2. Energy behanour of APF
v. POWR REQUERIMENTS ON THE ACTIVE POWER FILTER
In order to establish characteristics of the controller based
on the energy stored in the dc-bus, it is necessary to know the
power developed by the converter in an active filtering
application. In this sense, power requirements associated to a
generic load will be characterized, and later on, the
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components of this power that should be contributed by the
active filter will be evaluated. To generalize this study, utility
voltages will be supposed unbalanced, that is, utility voltages
present positive, negative and zero sequence components.
Therefore, load voltage in a-p-0 reference frame can be
expressed as
9)0 0
- cos(wt t *,)
where V,, , V., and V,, correspond to the rms voltage of
positive, negative and zero sequences respectively.
In this case, voltage imbalance in the point of common
coupling between the load and the utility is generally due to
voltage drops across the impedance of the generation,
transmission and distribution systems. Moreover, unbalancedcurrents circulation through machines, equipments and
conductors generates negative effects on the correct operation
conditions of these systems. Consequently, if minimization of
voltage imbalance in the point of common coupling is adopted
as an objective, the active power filter should assure that
current drawn from the source is balanced, sinusoidal at
fundamental frequency and in phase with the positive sequence
voltage of the utility. Also, the mean value of instantaneous
active power supplied by the source must coincide with the
mean value of instantaneous active power consumed by the
load.
For the study of power, it is firstly considered an unbalanced
linear load with connection to the neutral. Therefore, load
current can be expressed as:
sin(ot t J0,)
(10)0 0
- cos(o1 t J+,) cos(o1+ J.,)
Consequently, the mean value of instantaneous active power
consumed by the load, P L , + ,s:-
PE,( = P L + P L O = 3[V*J+,C W + , I ) +
(11)
V . , L , cos(#., -6,+ V 0 , LCOS(4 I - J I]where pLoand p L are the zero sequence power and real power
consumed by the load, respectively.
Tacking into account that characterization of load power
requirements is the objective of this study, intemal power
losses of the active filter will be intentionally neglected.
Therefore, the mean value of instantaneous active power
supplied by the source is given by (12).
To generate the active power given by 12), purely active
positive sequence currents in the source side should be:
With these currents, the whole instantaneous powers
supplied by the source should be:
[E\=[;]+[
[ *I [ i n ( 2 o t t j,, t
[;:I=FO PLO;I [ Pso h o
(14)0
= + PL O - (PL P d A c o s ( 2 o t + h,+ 4J .
In (14), oscillatory terms at 2 0 frequency appear as a
consequence of interaction between the positive sequencecurrent and the negative sequence voltage in the source side.
As (15) shows, instantaneous powers developed by the
active power filter are calculated by subtracting the powers
supplied by the source from the powers consumed by the load.
(15)
It is shown in (15) that the active filter is supplying the
whole load zero sequence power, by means of zero sequence
currents, p F o= i Lo+2Lo.f the mean value of this power
were different from zero, the filter would make a net transfer
of energy to the load. However, the filter is requesting, in theform of positive sequence sinusoidal currents, a mean value o f
real power equal to the mean value of zero sequence power
supplied to the load, FF = - P L O .Consequently, a net transfer
of energy will not exist in the filter, and the mean value of the
energy stored in the dc-bus will be constant, AZ* = 0 . The
filter also supplies, in the form o f positive and negative
sequence currents, the oscillating real power of the load, and it
consumes, in form of sinusoidal positive sequence currents, the
oscillating real power of the source, aF= p , - , These
oscillations at double frequency of utility frequency will
appear in the energy stored in the dc-bus. Finally, the filter
develops, in the form of sinusoidal positive and negative
sequence currents, whole imaginary power associated with the
load and the source, qF =qL F , -Fs. Logically, thisimaginary power developed by the filter will not generate
variation of energy stored in the dc-bus.
If the load is nonlinear, the load currents will contain
harmonic components of higher frequency in addition to the
components at fundamental frequency. Therefore, the active
power filter will inject, besides the currents at fundamental
frequency, the necessary curr ents . to compensate these
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harmonics. Then new high-frequency components will appear
in the real, imaginary and zero sequence powers shown in ( I S ) ,
which originate oscillations o f higher frequency in the energy
stored in the dc-bus.
If the utility voltage was not only unbalanced, but also
distorted, the situation would not differ excessively from that
one previously studied. In this case, additional oscillatingterms of power appear as a consequence of interaction between
voltages and currents of different frequency and/or sequence,
and mean value of instantaneous active power that should be
supplied by the source, j , in the form of purely active
positive sequence currents, would be:
+ V. L, cos(@-,-U QJ,, cos(d0, -so )],
where n indicates the harmonic order.
developed by the active filt er,p NC , s given as follow:
Therefore, by using (IS), the instantaneous active power
PP34 =PF+ P F O = PILO +P I , 5s = F L 3 ( P I s > (17)
where pIL3 is the oscillating active power consumed by the
load, and 5, is the oscillating real power associated to the
source. Fig. 3 shows a block diagram dedicated to obtaining
reference, currents for the active filter from calculation of the
mean value o f active power consum ed by the load.
LfVL 0 V+L
Fig. 3. Algorithm implementation for filter reference c m n l s determination
In Fig. 3, the transfer function of th e low-pass filter is:
Cut-off frequency of this filter has been chosen taking into
account that the minimum frequency of the oscillations in the
load active power can become equal to 2ws, being as he
fundamental utility frequency. Moreover, unitary damping
factor has been chosen to avoid over-peak in the step response
of filter.
From this study of powers, the block diagram shown in Fig.
4 represents energey solicitation on the active power filter.Note that noncompensated intemal power losses have been
included in this diagram.
VI. CONTROLLER OF I X - B U S ENERGY VARIATION
If the controller of dc-bus ,differential voltage is operative, it
is possible to accept that v C , = - v c I . and therefore energy
stored in the dc-bus can be calculated by:
19)I 2 1 2
-- CI - c 2 =- v, ,d= - 4 4
where C , = C, = C. Therefore, the algorithm used to estimate
energy variation in the dc-hus taking into account an initial
value adopted as a reference would be:
where Vdcc,efls the dc-b us vcsltage reference.
From Fig. 4, energy variation in dc-bus can be expressed as
AW,(S) = G(s)[HPF(s) .P, ,&)+ en, s)Ps(s )], 21)
where
s s t 20,)
(st0/ 2H P F ( s ) = ( l - L P F ( s ) ) = ( 2 2 )
If it is con sidered that en, s)- , s) = 0 , then the following
transfer function is obtained:
s t 20,
(23)wd' (s)= G( s ) HI ' F( s ) = - -
5 3 #
In (23). steady state value of step , response' is
Aw,.,, = -2 / 0, therefore, a suitable controller must be
designed to avoid this steady state error in the dc-bus energy
Th e proposed dc-bus energy controller is shown in Fig. 5 .
S(S)
Fig. 5. Control diagramof energy variation in the dc-bus
In Fig. 5 C s) is a proportional controller with a gain equal
to k, and H(s) is constituted by a notch filter in cascade with a
low pass filter tuned to q,=2rrg, and its transfer function is:
0, +0
H ( s )='( ' ') . 1 o - 10.0 I -2 .0 , . (24)
Characteristic transfer functions of the system depicted by
means of Fig. 5 are shown in (25 ) .
Fig. 4. Energy requirements for APF
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Since the whole oscillatory active power of the load at
w 2 q hould be developed by the active filter, the transfer
function shown in (25a) should have unitary gain at this
frequency. Thereby, proportional gain is limited to k = q . W ith
this value fork , and supposing that the H(s) block has not been
included in the controller, active power consumed by the load
at w2a3 would be slightly lagged respect to active power
supplied by the active filter. This would imply incorrect
compensation for this component of instantaneous active
power of the load by the active filter. To avoid this problem,
the H(s) block is added to the control loop. Fig. 7 shows
temporal evolution of the dc-bus energy variation when a
unitary step appears in the instantaneous active power
consumed by the load. The values ~ 2 . ~ 5 0 ,=rg-=2.n.10,
q = 2 ~00and tr= I have been considered.
,.ID
I
Fig. 7. Step response of the closed-loop system
An analytical study of (25h) permits to obtain a very
accurate approximate expression for the minimum value of the
waveform plotted in Fig. 7. This expression is shown in (26).
From (ZO), it is possible to determine the appropriate value
for the dc-bus capacitors, fixing as a design condition that
when a certain step appears in the average value of active
power consumed by the load, dp,, he dc-bus voltage does notdecrease below a limit value, Then, the appropria te
value for the capacitors is given by:
4AP,(I+
So far <,,,(s)-&(s) = 0 bas been considered. If now these
powers are kept in mind, an analysis of the diagram in Fig. 5
allows obtaining expression (28) for dc-bus energy variation.
AW,(s )=1-G s)C s)H s)
.[H ~ F ( s ) P , , , ( s ) + ~ ~ , ( s ) - ~ ~ ( s ~ ]
As Fig. 8 shows, to achieve a unique transfer function that
relates dc-bus energy variation with all powers terms shown in
(28). a second control loop is added to the system. Fig. 8
shows real structure of control diagram of dc-bus energy
variation. In this diagram, real structure of high-pass filter has
been considered, D(s)=s-is a derivative block, LPF(s) is the
low-pass filter shown in (IS), and a new variable for the
effective power that m odifies dc-bus energy has been defined
as PF d, s). nalysis of diagram shown in Fig. 8 permits to
obtain the following transfer function:
Expression (29) denotes that dc-bus energy variation is
completely controlled, since it follows an identical evolution
versus variations of the powers that affect the energetic state of
the filter. Analysing the control diagram shown in Fig. 8 t is
obtained that:
L1 =-( S ) H ( s ) C ( s ) +D( s ) L PF( s )
= - [ H ( S ) C ( S ) ~ ( ~ ) + F , ( S ) ] ,
AW,(s) I -LPF s) (30)
where F,(s) and F, s) are defined in (3 1
m C(S)
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Expression (30) leads to the definitive control diagramshown in Fig. 9.
PLJ 6) S ( S )
Fig. 9. Definitive dc-bus energy controller
This definitive control diagram has been implemented as
Fig. 10 shows. The reference current for leg d is obtained by
means of iid = - iia +jib iic , and power to current
transformation is given by:
Fig. IO . hplementation ofAPF controller
VII. SIMULATION RESULTS
, , .-2 ’
0 dW .
10 ........... . . . . . _ 1 ..........-20 - - . ..,..... ._ ......
7 - - ~~
0 0.05 0.1 0.15 0.2 0.25 0.3
t kl
Fig. I I . Simvlaled waveforms
VIII. CONCLUSION
From the study presented,, t can be concluded that proposed
controller provides an effective method for reference currents
calculation based on energy state of the active power filter.
With this approach, the dc-bus capacitors are rated according
to dynamic response of the system, so operation conditions of
the filter can be as sured for :sudden variations in load power.
REFERENCES
[ I ] C.A. Quinn and N. Mohan, “Active Filtering Currents in Three-phase,Four-Wire Systems with ThreePhasse and Single-phase Non-LinearLaads.” in Proe. IEEE- APEC, Feb. 1992, Boston, USA pp. 829-836.
S . Bum, L. Malesani, and. P. Mattavelli, ‘Comparison of Current
Control Techniques for Active filter Applications,” IEEE Trans.InduslriolEleelronics, vol. 45, no. 5. Oct. 1998, pp. 722-729.P. Verdelha, and G.D. Marques, “Four-wire Current-Regulated PWM
Voltage Converter,” IEEE Zions. Industrial Electronics, vol. 45, no. 5,
Oct. 1998,pp. 761-770.[ 4 ] R. Zhang, V.H. Prasad, D. Boroyevich and F.C. Lec, ‘Three-
Dimensional Space Vector Modulation for Four-Leg Voltage-Source
Converters.” IEEE Tram. Power Elecrronics. vol. 17. no.3. May 2002.
21
[3]
For simulation numoses. unbalanced utilitv voltaee has been DO 314-326
balanced resistive load is connected to the utility Later, at 4 , pp . 2 9 3 9 . 2 9 ~ .
f=O,15 s, another 2 kW nonlinear load is also connected. Q.-C. Zhong, T.C. Green, 1. Liang and G. Weiss, “H‘ Control of the
Neutral Point in 3-phase 4-Wire DC-AC Converters.” in Prae. IEEE-IECON , Sevilla, SPAIN, Nwr. 2002 vol. I , pp. 520-525.
analysis. Proper sinusoidal currents on the source side and a M,K, M ~ ~ ~ ~ ,, ,ashi and Ghosh, -cOntmlchemes for
minimum value of dc-bus energy variation according to that Equalization of Capacitor Voltages in Neutral Clamped Shunt
expected from ( 2 6 ) are obtained, Hence, &.bus energy control Compensator,” IEEE Tram. Power De l i v e y , vol. 18 no. , April 2003,
pp. 538-544.and load current co nditioning are perfectly achieved.
[ 6 ]
in ~ i ~ ,1, simulated corroborate the previous
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