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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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