Electrical Power and Energy Systems 61 (2014) 629–646

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Hybrid direct power/current control using feedback linearizationof three-level four-leg voltage source shunt active power filter

http://dx.doi.org/10.1016/j.ijepes.2014.03.0710142-0615/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Département de l’Electronique et des Communica-tions, Faculté des Nouvelles Technologies d’Information et Communication, Uni-versité Kasdi Merbah, Ouargla 30000, Algeria. Tel.: +213 661352139.

E-mail addresses: bouzidi.m.28@gmail.com (M. Bouzidi), aek_benaissa@yahoo.fr(A. Benaissa), sa_barkati@yahoo.fr (S. Barkat).

M. Bouzidi a,b,⇑, A. Benaissa b, S. Barkat c

a Département de l’Electronique et des Communications, Faculté des Nouvelles Technologies d’Information et Communication, Université Kasdi Merbah, Ouargla 30000, Algeriab Département d’Electrotechnique, Faculté des Sciences de l’Ingénieur, Université Djillali Liabes de Sidi Bel Abbes, Sidi Bel Abbes 22000, Algeriac Département d’Electrotechnique, Faculté de Technologies, Université de M’sila, M’sila 28000, Algeria

a r t i c l e i n f o

Article history:Received 3 January 2013Received in revised form 27 March 2014Accepted 28 March 2014Available online 3 May 2014

Keywords:Three-level four-leg shunt active filterHybrid direct power/current controlFeedback linearizationThree dimensional space vector modulation

a b s t r a c t

This paper proposes a hybrid direct power/current control-three dimensional space vector modulationcombined with feedback linearization control for three-phase three-level four-leg shunt active power fil-ter (SAPF). The four-leg SAPF ensures full compensation of harmonic phase currents, harmonic neutralcurrent, reactive power and unbalanced nonlinear load currents. It also regulates its self-sustaining DCbus voltage. The voltage-balancing control of two split DC capacitors of the three-level four-leg SAPF isachieved using three-level three dimensional space vector modulation with balancing strategy basedon the effective use of the redundant switching states of the inverter voltage vectors. Complete simula-tion of the resultant active filtering system validates the efficiency of the proposed nonlinear controlmethod. Compared to the traditional control, the use of feedback linearization control allows to exhibitexcellent transient response during balanced and unbalanced load, and grid voltage.

� 2014 Elsevier Ltd. All rights reserved.

Introduction

The excessive use of power electronic equipments, which repre-sent nonlinear loads, in a distribution network has caused manydisturbances in the quality of power such as harmonic pollutions,unbalanced load currents, and reactive power problems. As a resultpoor power factor, weakening efficiency, overheating of motorsand transformers, malfunction of sensitive devices etc. are encoun-tered [1–3].

Conventionally, a passive power filter which consists of passiveelements is used to provide harmonic filtering as an economicaland effective filtering device. However it has shortcomings suchas fixed compensation performance, bulk in size and resonancetroubles [4–7]. Important kinds of passive power filters and theirconfigurations are discussed in [4]. To overcome the shortcomingsof passive power filters and to mitigate the power pollution in net-works caused by the nonlinear loads, an active power filter (APF)was established in around 1970s [8–10]. APFs are previously notimplemented in power networks, because of unavailability of highspeed power switching devices. Recently the power electronic

development spurred the interest in IGBTs, MOSFETs, etc. [8] andthen APFs are developed incorporating power electronics technol-ogy to support the needs of industry. Shunt, series, and hybrid con-figuration are the three main types of three-phase, three-wireactive power filters and their merits and demerits are discussedin [4].

For medium to high power applications the multilevelconverters are the most attractive technology. Indeed, multilevelconverters have shown some significant advantages over tradi-tional two-level converters [9–12]. The main advantages of themultilevel converter are a smaller output voltage step, lower har-monic components, a better electromagnetic compatibility, andlower switching losses [9–12]. In the recent time, the use of multi-level inverters is prevailing in medium-voltage active power filterswithout using a coupling transformer [13–17].

In several areas, power is distributed through three-phase four-wire system and traditional APF is inadequate for harmonics com-pensation and power factor correction. To overcome this shortage,a three-phase four-wire SAPF has been introduced in the 1980s[18–22].

Basically there are two main kinds of three-phase, four-wireSAPF depending on their connection to the neutral wire. In the firstkind the neutral wire is connected to the midpoint of the DC-linkcapacitors. In [23,24], this approach was studied where the inver-ter was operating as an active power filter. However, although it issimple in terms of topology, this approach is not suitable for SAPF

630 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

application, for the following reasons [25,26]: (1) insufficient DC-link utilization, (2) high ripple on DC-link capacitors, (3) problemof DC-link capacitor voltages balance. In the second type the neu-tral wire is connected to the additional fourth leg, this topology hasbeen shown to be a solution for inverters operating in three-phasefour-wire systems and it offers full utilization of the DC-linkvoltage and lower stress on the DC-link capacitors [26].

Various control strategies have been proposed to control grid-connected DC/AC converters, a classic control usually based on gridvoltage [27,28] or virtual-flux [29] oriented vector control (VOC orVFOC) scheme. This scheme decomposes the AC current into activeand reactive power components in the synchronously rotatingreference frame. Decoupled control of instantaneous active andreactive powers is then achieved by regulating the decomposedconverter currents using proportional integral (PI) controllers.One main drawback for this control method is that the perfor-mance highly relies on the completeness of current decoupling,the accurate tuning of PI parameters, and the connected grid volt-age conditions.

Based on the principles of the well-known direct torque control(DTC) [30,31] of AC machines, an alternative control approach,namely, direct power control (DPC) was developed for the controlof grid-connected voltage-sourced converters [32,33]. Similar tothe traditional DTC, lookup table direct power control (DPC-LUT),as the name indicates, selects the proper converter switching sig-nals directly from an optimal switching table on the basis of theinstantaneous errors of active and reactive powers, and the angularposition of converter grid voltage [33] or the virtual flux vector.This later result from the integration of converter grid voltage mea-sured with voltage transducers [34] or estimated based on the DC-link and the converter switch states [34]. The main disadvantage ofDPC-LUT is the variation of switching frequency, which generatesan undesired broadband harmonic spectrum range and makes itpretty hard to design a line filter.

These disadvantages can be effectively overcome by using spacevector modulation (SVM) algorithm to replace the traditionalswitching table. The combination of SVM and traditional DPC formsthe space vector modulation direct power control (SVM–DPC) [35].

Indeed the traditional two dimensional SVM algorithms onlycan be used to control converter connected to power system withbalanced voltage/current where the homopolar component in Con-cordia transformation is equal to zero. In the four-wire systemdistribution, the case of unbalanced voltage is taken in consider-ation; therefore, the homopolar component is not equal to zero.Thus, three dimensional space vector modulation (3DSVM) algo-rithms must be taken into account in order to generate the desiredsignal.

In [25,26], 3DSVM schemes are analyzed for a four-leg two-level voltage source inverter. Authors in [23,24] have presented amodulation scheme for a three-level inverter as an active powerfilter in three-phase four-wire systems where hysteresis modula-tion was designed in a three-dimensional domain. However,3DSVM for a four-leg three-level inverter has not yet been studiedin stationary reference frame. A novel algorithm of space vectormodulation for a four-leg three-level inverter is proposed in thispaper. The effectiveness of the proposed modulation algorithm,and the advantages of the proposed topology over conventionalones, are discussed and verified with simulation results with SAPFsystem.

Commonly, the abovementioned control techniques are basedon traditional PI control. In order to improve the performance ofthree-phase four-leg SAPF, various nonlinear control strategieshave been reported in the literature. The proposed control strate-gies include among others sliding mode control [36], passivity con-trol [37], nonlinear optimal predictive control [38] and H1 control[39].

In this paper, a nonlinear control strategy based on the feedbacklinearization associated to hybrid direct power/current controlwith 3DSVM (DP/CC-3DSVM) is applied to three-phase three-levelfour-leg SAPF in order to improve its performances. It is wellknown that feedback linearization technique is a control methodwhich aims to eliminate the nonlinearity of the system by usingthe inverse dynamics [40]. It has been applied successfully to con-trol the three-level neutral point clamped (NPC) boost converter[41], three-phase AC/DC PWM converters [42] and three-levelthree-phase shunt active power filter [43]. As a result, good perfor-mances have been reported and the nonlinear control law was ableto reduce the influence of parametric variations, utility distur-bances, and DC load shedding [41].

This paper is organized as follows. In Section ‘Four-leg shuntactive power filter’, the configuration of four-leg SAPF is presentedand the system model is developed. In Section ‘Control of three-level four-leg compensator’, the feedback linearization associatedto the DP/CC-3DSVM is investigated. The nonlinear controllersare synthesized and the three-level 3DSVM with balancing capabil-ity are presented also in this section. In Section ‘Simulation results’,the performances of controlled system are verified by simulationresults. Finally, in Section ‘Conclusion’ some conclusions areestablished.

Four-leg shunt active power filter

System description

The basic compensation principle of the four-leg SAPF is shownin Fig. 1. The main task of the four-leg SAPF is to reduce harmoniccurrents and to ensure reactive power compensation. Ideally, thefour-leg SAPF needs to generate just enough reactive and harmoniccurrent to compensate the nonlinear load harmonic in the line. Theresulting total current drawn from the AC main is sinusoidal andbalanced. The compensated neutral current is provided through afourth leg allowing a better controllability than the three-leg withsplit-capacitor configuration. The main advantage of the four-legconfiguration is the ability to suppress the neutral current fromthe source without any drawback in the filtering performance.

Mathematical model of the three-level four-leg SAPF

The switching functions are defined as Fij where i e {a, b, c, n} isthe phase and j e {0, 1, 2}is the voltage level. Fij takes value ‘‘1’’ if i-phase is connected to voltage level j and ‘‘0’’ otherwise; theseswitching functions can be expressed as:

Fx2 ¼ Sx2Sx1

Fx1 ¼ Sx2�Sx1 x ¼ a; b; c or n

Fx0 ¼ �Sx2�Sx1

ð1Þ

The instantaneous AC converter phase to neutral voltages vFa,vFb and vFc can be expressed in terms of switching functions andDC-link voltages capacitors as given by:

vFa

vFb

vFc

264

375 ¼

Fa2 � Fn2 Fa1 � Fn1 Fa0 � Fn0

Fb2 � Fn2 Fb1 � Fn1 Fc0 � Fn0

Fc2 � Fn2 Fc1 � Fn1 Fc0 � Fn0

264

375

vC2 þ vC1

vC1

0

264

375 ð2Þ

The mathematical equations which govern the behavior of theAC-side of SAPF are:diFa

dt¼ 1

LFðvFa � va � RFiFaÞ

diFb

dt¼ 1

LFðvFb � vb � RFiFbÞ

diFc

dt¼ 1

LFðvFc � vc � RFiFcÞ

ð3Þ

0i

1i

Fav

scv

sbv

sav

av

bv

cv

FaiFbiFciFni

FcR FcL

FbLFbR

FaR FaL

1Cv

2Cv

2i

1C

2C

2Ci

1Cidcv

1aS

2aS

1aS

2aS

1bS

2bS

1cS

2cS

1nS

2nS

1bS

2bS

1cS

2cS

1nS

2nS

Nonlinear

Load

sai

sbi

sci

sni

Lai

Lbi

Lci

Lni

saR saL

sbR sbL

scR scL

LaR LaL

LbR LbL

LnR LcL

Fbv

Fcv

PCC

Fig. 1. Three-level four-leg SAPF configuration.

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 631

where va; vb and vc are the point of common coupling (PCC) volt-ages, iFa; iFb and iFc;vFa; vFb and vFc represent AC side currents andvoltages of the SAPF, respectively.

Assuming C1 = C2 = C, the DC side of the filter can be expressed as:

dvdc

dt¼ i1 � i2

Cð4Þ

i1 and i2 are the DC-side intermediate branch currents.

3 33

2

22

*dcp

*Fo Loi i=

Lq

cv

bv

av

v αβ

sav

sbv

scv

Foi

Lp

Lp

Fi αβFoi

F Fp q

FabciLabci

Li αβ Loi

sci

sbi

sai

Fai

Fni

Fbi

Fci

abcv

sni

αβabc abc

oαβabc

oαβ

sR sL

sR sL

sR sL

2

~

Fig. 2. Nonlinear control scheme of direct power/

Assuming the modulation algorithm will balance voltages incapacitors (vC1 = vC2 = vdc/2), the DC side dynamic equation can bewritten as follows:

dvdc

dt¼ pdc

Ceqvdcð5Þ

where pdc is the DC active power, and Ceq = C/2.

8

Lni

*Fv α

*Fv β

*Fov

Fav

Fbv

Fcv

1Cv

2Cv

2Cv

1C

2C

Lci

Lbi

Lai

1lL1lR

abcnS

dcv

*dcv

FR FL

FR FL

FR FL

2lL2lR

3lL3lR

*dcp

current control for three-level four-leg SAPF.

2000

1101

2002

2100

1201

2201

1200

0201

2101

2102

2200

2202

1202

0200

0202

0210

0211

0212

0220

0221

0222

0120

0121

0122

0020

0021

0022

1020

1021

1022

2020

2021

2022

2010

2101

2012

2110

2111

2112

1000

1001

1002

2210

2211

2212

1100

1102

1210

1211

1212

0100

0101

0102

1220

1221

0201

0110

0111

0112

1120

1121

1122

0010

0011

0012

2120

2121

2122

1010

1011

1012

2001

2220

2221

2222

1110

1111

1112

0000

0001

0002 3 dcv−

5

2 3dcv−

2

3dcv−

32

dcv−

3dcv−

2 3dcv−

2 3dcv

β3

dcv

32

dcv

2

3dcv

5

2 3dcv

3 dcv

o

α

Prism 3 Prism 2 Prism 1Prism 4

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

7

6

5

4

3

2

1 1

2

3

4

5

6

7

22202210

2110

2221

2210

2110

222122111100

1110

1110

2110

22211110 2211

1100

21111000

21111000

22211110 2211

1100

222211110000

22111100

222211110000

11012212

21111000

21111000

222211110000 1101

2212

21121001

11012212

222211110000

21121001

11120001

11012212

21121001

11120001

1102

1102

21121001

11120001

1002

1002

110211120001

0002

2210

2200

2100

22111100

2200

2100

22111100 2201

2201

2100

22111100

21012201

2101

22111100

11012212

2201

2101

11012212 2202

2202

2101

11012212

2102

2102

220211012212

1102

2210

2110

2100

22111100

22111100

21002110

21111000

2100

22111100

21111000 2101

22111100

21111000

2101

11012212 2111

1000 2101

11012212

21121001

11012212

2101

21121001

2102

11012212

210221121001

1102 21121001

1102

2102

1002

2110

2100

2000

21111000

21111000

2000

2100

2101 200021111000

2101

2001

2001

2101

21111000

21121001

2101

2001

21121001

2102 2001

2102

21121001

2002

2002

2102

21121001

1002

10 Tetrahedrons

7 Tetrahedrons

8 Tetrahedrons

7 Tetrahedrons

(a) (b)

Fig. 3. Three dimensional representation, (a) Switching voltages vectors in abo coordinates, (b) tetrahedrons in the first sector.

632 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 633

Based on the Concordia coordinates transformation, the differ-ential equations describing the dynamic model of the four-leg SAPFin abo reference frame are given by (6)

diFa

dt¼ 1

LFðvFa � va � RFiFaÞ

diFb

dt¼ 1

LFðvFb � vb � RF iFbÞ

diFo

dt¼ 1

LFðvFo � vo � RF iFoÞ

dvdc

dt¼ pdc

Ceqvdc

ð6Þ

-1500-1000

-5000

5001000

1500

-1000

-500

0

500

1000-500

0

500

vFα*

(a)

vFβ*

v Fo*

-1500 -1000 -500 0 500 1000 1500-1000

-800

-600

-400

-200

0

200

400

600

800

1000

vFα*

v Fβ*

(b)

Fig. 4. Trajectory of reference voltage vector under unbalanced sinusoid conditionwith conventional algorithm (a) trajectory in 3-dimensional space; (b) projection ofthe trajectory on ab plan.

Control of three-level four-leg compensator

The basic operation of the proposed control method is shown inFig. 2. The nonlinear loads are constructed from three uncontrolledsingle-phase rectifiers. The capacitor voltage is compared with itsreference value v�dc , in order to maintain the energy stored in thecapacitor constant. The proposed nonlinear controller is appliedto regulate the error between the capacitor voltage and itsreference. The output of nonlinear voltage controller presents thereference of DC active power p�dc. The compensating powers arecompute using the instantaneous p–q theory, and the referenceof the homopolar current i�Fois chosen equal the homopolarcomponent of the nonlinear load current [35]. The alternate valueof active power is extracted using high-pass filter (HPF). The out-put signals from nonlinear power and current controller are usedfor switching signals generation by a 3DSVM.

p–q Theory based control strategy

Instantaneous active and reactive powers of the nonlinear loadare computed as:

pL

qL

� �¼

va vb

�vb va

� �iLa

iLb

� �ð7Þ

The instantaneous active and reactive powers include AC andDC values and can be expressed as follows:

pL ¼ �pL þ ~pL

qL ¼ ~qL þ ~qLð8Þ

DC values ð�pL; �qLÞ of the pL and qL are the average active andreactive power originating from the positive-sequence componentof the nonlinear load current. AC values ð~pL; ~qLÞ of the pL and qL arethe ripple active and reactive powers [44].

For harmonic, reactive power compensation and balancing ofunbalanced three-phase load currents, all of the reactive power(�qL and ~qL components) and harmonic component ~pL of active powerare selected as compensation power references as follows [44]:

p�Fq�F

� �¼

~pL � pdc

qL

� �ð9Þ

The signal pdc is used as an average real power and is obtainedfrom the nonlinear DC voltage controller.

Since the zero-sequence current must be compensated the ref-erence of homopolar current is given as:

i�Fo ¼ iLo ð10Þ

Feedback linearization control

Model subdivisionThere are four outputs to be controlled: DC capacitor voltage

vdc, homopolar current component iFo, active power pF, and reactive

power qF. In order to ensure that each of the previously mentionedoutputs follows its reference (v�dc; i

�Fo; p

�F andq�F respectively), it is

appropriate to divide the system (6) into two subsystems:Subsystem 1:The first subsystem is described by the following equation:

diFa

dt¼ 1

LFðvFa � va � RFiFaÞ

diFb

dt¼ 1

LFðvFb � vb � RFiFbÞ

diFo

dt¼ 1

LFðvFo � vo � RFiFoÞ

ð11Þ

When the first subsystem is expressed in the form of (12):

_x ¼ f ðxÞ þ gðxÞu ð12Þ

It results:

f ðxÞ ¼f1

f2

f2

264

375 ¼

� RFLF

x1 � 1LF

vFa

� RFLF

x2 � 1LF

vFb

� RFLF

x3 � 1LF

vFo

2664

3775

gðxÞ ¼

1LF

0 0

0 1LF

0

0 0 1LF

2664

3775

Table 1Prism identification in each sector k (k = 1or 2).

lk1 lk2 PRki

0 0 PRk4ðifU�kF1 þ U�kF2 < 1Þ

PRk2ðifU�kF1 þ U�kF2 P 1Þ

1 0 PRk1

0 1 PRk3

Where PRki is a prism number i located in sector k.

24PR *1

2FU β

22PR

*FU

*FU α

abcx

ψ*1

1FU α

14PR

222x111x000x

210x

200x211x100x

121x010x

221x110x

220x120x020x

1st Sector

2nd Sector

β

13PR

α

11PR

12PR

*FU β

21PR2

3PR

Fig. 6. Space voltage vectors for a three-level four-leg inverter in sector one andtwo.

-5000

5001000

1500

0

500

1000

1500-500

0

500

UFα

*

(a)

UFβ*

UF

o*

-600 -400 -200 0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

UFα*

UF

o*

(b)

Fig. 5. Trajectory of reference voltage vector under unbalanced sinusoid conditionwith proposed algorithm (a) trajectory in 3-dimensional space; (b) projection of thetrajectory on ab plan.

634 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

x ¼x1

x2

x3

264

375 ¼

iFa

iFb

iFo

264

375;u ¼

u1

u2

u3

264

375 ¼

vFa

vFb

vFo

264

375

Subsystem 2:The second subsystem is defined by Eq. (5), which has only one

state x = vdc and only one control input u = pdc. The second subsys-tem can be also written in the form (12).

Table 2Interchanging the switching states in odd sectors.

Sector 1 Sector 3 Sector 5

a a ? b a ? cb b ? c b ? ac c ? a c ? bn n ? n n ? n

Table 3Interchanging the switching states in pair sectors.

Sector 2 Sector 4 Sector 6

a a ? b a ? cb b ? c b ? ac c ? a c ? bn n ? n n ? n

No NoYes Yes

*Fv

Yes No

*FU

1=2=

Fig. 7. Schematic diagram of the 3DSVM with proposed algorithm and balancing DCcapacitors voltages.

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 635

Where

f ðxÞ ¼ 0; gðxÞ ¼ 1Ceqvdc

0.3 0.305 0.31 0.315 0.32

-50

0

50

Time (s)

i sa

(A)

(a)

DC voltage controller synthesisThe synthesis of the DC voltage controller is based on the sec-

ond subsystem.The derivative of the output y = vdc is given by:

_y ¼ Lf hðxÞ þ LghðxÞu ð13Þ

where Lfh stands for the Lie derivative of h with respect to f, simi-larly Lgh.Then:

_y ¼ pdc

Ceqvdcð14Þ

The control input pdc appears in (14), so the relative degree isr = 1. The relative degree of this output is equal to the order of sub-system 2, which corresponds clearly to an exact linearization [40].

Then the control law is obtained by:

p�dc ¼ Ceqvdcv ð15Þ

where

_y ¼ v ð16Þ

For a problem of tracking of trajectory defines by v�dcðtÞ, theterm v is expressed by:

Table 4System parameters.

RMS value of phase voltage 220 V

DC-link capacitor C1, C2 5 mFSource impedance Rs, Ls 0.01 mX, 1 mHFilter impedance RF, LF 0.01 mX, 1 mHLine impedance RL, LL 0.01 mX, 1 mHDC-link voltage reference v�dc 800 VDiode rectifier load Rl , Ll 5 X, 10 mHSwitching frequency fs 5 kHzSampling frequency 1 MHzk, k1 = k2 = k3 constants 50, 8 � 105

0.3 0.305 0.31 0.315 0.32

-50

0

50

Time (s)

i sa

(A)

(a)

(b)

Fig. 8. (a) Source current before harmonics compensation, (b) Its harmonicspectrum.

v ¼ kðv�dc � vdcÞ þdv�dc

dtð17Þ

where k is a positive constant.

Power and current controller synthesisThe outputs of the first subsystem are selected as:

y1 ¼ pF

y2 ¼ qF

y3 ¼ iFo

ð18Þ

And their derivatives are given by:

ddt

y1 y2 y3½ �T ¼ fðxÞ þ DðxÞu ð19Þ

(b)

Fig. 9. (a) Source current after harmonics compensation using nonlinear controller,(b) Its harmonic spectrum.

where

0.4 0.405 0.41 0.415

-50

0

50

Time (s)

i sa (

A)

(a)

(b)

Fig. 10. (a) Source current after harmonics compensation using PI controller, (b) Itsharmonic spectrum.

636 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

fðxÞ ¼

vaf1 þ vbf2

vbf1 � vaf2

f3

2664

3775; DðxÞ ¼

vaLF

vb

LF0

vb

LF� va

LF0

0 0 1LF

26664

37775

The control inputs appear in (19). In this case, the relativedegree is r = r1 + r2 + r3 = 3. The relative degree of the chosen

0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24

-50

0

50

0 0.1 0.2 0.3 0-100

0

100

Tim

Loa

d cu

rren

ts i

Lab

c (A

)(b)

0.46 0.47 0.48 0.49 0.

-100

-50

0

50

100

0 0.1 0.2 0.3 0

-50

0

50

Tim

SAP

F c

ourr

ents

iF

abcn

(A)(c)

iFa

iFb

iFc

iFn

Without SAPF Balanced load

0 0.1 0.2 0.3 0

-50

0

50

Tim

Sour

ce c

urre

nts

i sab

c (A

)

(a)

Zoom

0.46 0.47 0.48 0.49 0.

-50

0

50

Fig. 11. Simulation results of the proposed D

outputs is equal to the order of subsystem 1, than it is about ofan exact linearization [40].

The decoupling matrix determinant is different to zero, andthen the control law is given as:

u ¼u1

u2

u3

264

375 ¼ DðxÞ�1 �fðxÞ þ

v1

v2

v3

264

375

264

375 ð20Þ

.4 0.5 0.6 0.7 0.8e (s)

5 0.51 0.52 0.53 0.54

.4 0.5 0.6 0.7 0.8e (s)

With SAPF Unbalanced load

.4 0.5 0.6 0.7 0.8e (s)

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54

-50

0

50

Zoom

Zoom

5 0.51 0.52 0.53 0.54

Zoom

P/CC-3DSVM using nonlinear controller.

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 637

The application of the linearization law on the first subsystemled to two decoupled linear systems, given by:

_y1

_y2

_y3

264

375 ¼

v1

v2

v3

264

375 ð21Þ

0 0.1 0.2 0.3

-20

0

20

T

Neu

tral

cur

rent

isn

(A

)(d)

0.16 0.17 0.18 0.19

-20

-10

0

10

20

0 0.1 0.2 0.3

-1

0

1

2

3

x 10 4

T

Sour

ce p

ower

ps

(W)

and

q s(V

ar)

(e)

0 0.1 0.2 0.3700

750

800

850

900

T

DC

vol

tage

and

its

re

fere

nce

(V)

(f)

0 0.1 0.2 0.3350

400

450

T

DC

cap

acit

ors

vol

tage

s (V

)

(g)

0.3 0.31 0.32

-200

0

200

T

v a (

V)

and

i sa

(A)

(h)

Zoom

Fig. 11 (con

The control law used for tracking is:

v1 ¼ k1ðp�F � pFÞ þdp�Fdt

v2 ¼ k2ðq�F � qFÞ þdq�Fdt

v3 ¼ k3ði�Fo � iFoÞ þ di�Fodt

ð22Þ

where k1, k2 and k3 are positive constants.

0.4 0.5 0.6 0.7 0.8

ime (s)

0.2 0.21 0.22 0.23 0.24

0.4 0.5 0.6 0.7 0.8

ime (s)

0.4 0.5 0.6 0.7 0.8ime (s)

0.4 0.5 0.6 0.7 0.8

ime (s)

0.33 0.34 0.35 0.36

ime (s)

tinued)

638 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

Three dimensional space vector modulation

In a three-level three-legged converter, there are 33 possibleswitch combinations. With the fourth neutral leg, the total numberof combinations is 34. The switch combinations are represented byordered sets (SaSbScSn).

Where

Sa ¼ 2 if Fa2 ¼ 1Sa ¼ 1 if Fa1 ¼ 1Sa ¼ 0 if Fa0 ¼ 1

8><>: ð23Þ

The same notation applies to phase legs b and c and the fourthneutral leg.

For switching combinations, the vectors given by transforming(2) to abo coordinates, can be described using a graphical represen-tation in three-dimensional space as shown in Fig. 3a.

There are three zero switching vectors (2222, 1111, 0000), and65 unique non-zero switching vectors. It can be viewed as thateach of the switching vectors for a three-legged converter splitsinto three switching vectors, depending on switch position of theneutral leg. All the 81 switching vectors can be sorted into thirteenlayers.

The diagram of space vectors can be divided into six sectorswith every sector further divided into four prisms. As shown inFig. 3b, the prisms 1 and 3 are formed by 7 tetrahedrons, whilethe prisms 2 and 4 are formed by 8 and 10 tetrahedronsrespectively.

In each tetrahedron, three tasks must be done: localization ofthe reference voltage vector v�F , calculation of duration time inter-vals of adjacent switching vectors, and the generation of the corre-sponding pulses. This increases the required computational timeand augments the hardware and software complexity.

0 0.1 0.2 0.3-1000

0

1000

Ti

SAP

F v

olta

ge V

Fa

(V)

(i)

0 0.1 0.2 0.3-1000

0

1000

Ti

SAP

F v

olta

ge V

Fb (

V)

(j)

0 0.1 0.2 0.3-1000

0

1000

Ti

SAP

F v

olta

ge V

Fc (

V)

(k)

Fig. 11 (con

Proposed 3DSVM algorithmThe proposed algorithm can reduce remarkably the complexity

of 3DSVM by using two sectors only (sector one and sector two) inthe conception of all modulation algorithm steps such as: determi-nation of the space vector location, duration time calculation, andpulses generation.

The six sectors are divided into two identical groups, odd sec-tors (1, 3 and 5) and pair sectors (2, 4 and 6).

The reference voltage vector usually turns in the space abo andcrosses all the sectors (Fig. 4), then, it is necessary to build anothervector U�F which turns only in sectors 1 and 2 and takes all informa-tion about v�F in the other sectors, as shown in Fig. 5.

The components of new reference voltage vector are:

U�Fa ¼ U�Fab cosðwÞU�Fb ¼ U�Fab sinðwÞU�Fo ¼ v�Fo

ð24Þ

where U�Fab ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�2Fa þ v�2Fb

qand w = mod(h, 2p/3) mod(x,y): is a func-

tion which gives the division remainder of x on y, w and h are theangles of the vectors U�F and v�F projected in ab plane respectively.The angle h given by:

h ¼ tan 2�1 v�Fb

v�Fa

� �ð25Þ

And: tan2�1: is a function returns the four-quadrant inversetangent.

Determination of the space vector location. The space vector locationis determined in three steps: (1) determining the sector number ofwhere the vector lies, (2) determining the number of prisms, and(3) determining the tetrahedron number of where the referencevector located.

0.4 0.5 0.6 0.7 0.8

me (s)

0.4 0.5 0.6 0.7 0.8

me (s)

0.4 0.5 0.6 0.7 0.8

me (s)

tinued)

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 639

Steps one and two are similar to those of three-level 2DSVM[45,46].

Step 1: Sector number computation.

The sector numbers are given by [46]:

s ¼ ceilh

p=3

� �2 f1;2;3;4;5;6g ð26Þ

0 0.1 0.2 0.3

-50

0

50

Tim

Sour

ce c

urre

nts

i sab

c (A

)(a)

0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24

-50

0

50

0 0.1 0.2 0.3

-100

0

100

Ti

Loa

d cu

rren

ts i

Lab

c (A

)(b)

0.46 0.47 0.48 0.49 0

-100

-50

0

50

100

Zoom

Z

0 0.1 0.2 0.3

-50

0

50

Tim

SAP

F c

ourr

ents

iF

abcn

(A)(c)

iFa

iFb

iFc

iFn

0.46 0.47 0.48 0.49 0-100

-50

0

50

100

Fig. 12. Simulation results of the propose

where ceil is the C-function that adjusts any real number to thenearest, but higher, integer.

Step 2: Prisms identificationReference vector U�F is projected on the axes of 60� coordinate

system [45]. In sector one and two, the normalized projected com-ponents are U�kF1 and U�kF2 given by (27) (k e {1, 2} for sector one ortwo). Fig. 6 shows the projection of U�F in the first sector.

0.4 0.5 0.6 0.7 0.8

e (s)

0.4 0.5 0.6 0.7 0.8

me (s)

.5 0.51 0.52 0.53 0.54

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54

-50

0

50

Zoom

oom

0.4 0.5 0.6 0.7 0.8

e (s)

.5 0.51 0.52 0.53 0.54

Zoom

d DP/CC-3DSVM using PI controller.

640 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

U�kF1 ¼U�Fab cosðw� ðk� 1Þ p3ÞÞ �

U�Fabffiffi3p sinðw� ðk� 1Þ p3Þffiffi

23

qvdc2

U�kF2 ¼2ffiffi3p U�Fab sinðw� ðk� 1Þ p3Þffiffi

23

qvdc2

ð27Þ

0 0.1 0.2 0.3-40

-20

0

20

40

Nat

ural

cur

rent

isn

(A

)(d)

0.16 0.17 0.18 0.19-40

-20

0

20

40

0 0.1 0.2 0.3

-2

0

2

4x 10 4

Sour

ce p

ower

s p s

(W)

and

q s(Var

)

(e)

0 0.1 0.2 0.3700

750

800

850

900

DC

vol

tage

and

its

re

fere

nce

(V)

(f)

0 0.1 0.2 0.3350

400

450

DC

cap

acit

ors

vol

tage

s (V

)

(g)

Zoom

0.3 0.31 0.32-400

-200

0

200

400

v sa (

V)

and

i sa (

A)(h)

Fig. 12 (con

In order to identify the prism where the required reference volt-age vector is located, the following integers are used:

lk1 ¼ intðU�kF1Þ

lk2 ¼ intðU�kF2Þ

ð28Þ

0.4 0.5 0.6 0.7 0.8

Time (s)

0.2 0.21 0.22 0.23 0.24

0.4 0.5 0.6 0.7 0.8

Time (s)

0.4 0.5 0.6 0.7 0.8

Time (s)

0.4 0.5 0.6 0.7 0.8

Time (s)

vC2

vC1

0.33 0.34 0.35 0.36

Time (s)

tinued)

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 641

where the int() function returns the nearest integer that is less thanor equal to its argument. The prism number is obtained according tothe value of lk

1 and lk2, as shown in Table 1.

Step 3: Tetrahedron identificationAfter the selection of prism, the next step is to determine the

tetrahedron according to the location of the reference voltage. Asshown in Fig. 3, each tetrahedron is limited from the top andthe bottom by two planes.

Each plane is created by three switching vectors. For example,the localization condition of tetrahedron 1 in the prism 1 of thefirst sector numbered as TeTk;i

j (k = 1 for sector 1, i = 1 for prism 1and j = 1 for tetrahedron 1) is given by:

U�Fo 6 �U�Fa þffiffiffi32

rvdc

U�Fo >U�Fa2þ

ffiffiffi3p

2U�Fb

ð29Þ

Duration time calculation. In order to minimize the circulatingenergy and to reduce the current ripple, switching vectors adjacentto the reference vector should be selected. At any sampling instantthe tip of the voltage vector lies in a tetrahedron formed by thefour switching vectors adjacent to it. The on-duration time inter-vals of each vector are obtained in accordance to the average valueprinciple, which is given by [45,46]:

v1t1 þ v2t2 þ v3t3 þ v4t4 ¼ U�FTs

t1 þ t2 þ t2 þ t4 ¼ Tsð30Þ

where Ts is the switching period, v1, v2, v3 and v4 are the fourswitching vectors adjacent to the reference voltage vector, andt1, t2, t3 and t4 are their calculated on-duration time intervalsrespectively.

Expression (30) can by decomposed in the abo coordinates sys-tem as follows:

0 0.1 0.2 0.3-1000

0

1000

SAP

F v

olta

ge V

Fa (

V)(i)

0 0.1 0.2 0.3-1000

0

1000

SAP

F v

olta

ge V

Fb (

V)(j)

0 0.1 0.2 0.3-1000

0

1000

SAP

F v

olta

ge V

Fc

(V)(k)

Fig. 12 (con

v1a v2a v3a v4a

v1b v2b v3b v4b

v1o v2o v3o v4o

1 1 1 1

26664

37775

t1

t2

t3

t4

26664

37775 ¼

U�FaTs

U�FbTs

U�FoTs

Ts

26664

37775 ð31Þ

With a proposed algorithm, the on duration time intervals are cal-culated only in sector one and sector two.

Pulse generation. The final step is to apply the calculated durationtime intervals to the corresponding vectors in each tetrahedron.The pulses are generated only in sector one and two, the other sec-tor can deduced by simply interchanging the states of the outputphases of pair and odd sectors as given in Table 2 (see Table 3).

In appendix’s table an equivalence between switching states ofeach tetrahedron located in same prism for sectors one and twocan be observed.

Based on this equivalence, the generation of the pulses in all tet-rahedrons located in same prism is not necessary. For examplethere are seven tetrahedrons in prism one located in sector one,the pulses are generated only in four tetrahedrons, as given inappendix’s table.

DC-capacitor voltages balancing based on minimum energy propertyIn the three-level three-leg three-wire NPC topology, the bal-

ancing problem of the DC-link voltage can be solved using theredundant vectors due to their effect on the DC-link capacitorsvoltage [45,46]. This paper shows clearly that the balancing prob-lem of the DC-link voltage in the three-level four-leg topology canbe solved using the redundant vectors in a similar way.

The electrical energy stored in the chain of DC-link capacitors isgiven by:

E ¼ C2ðv2

C1 þ v2C2Þ ð32Þ

When all capacitor voltages are balanced, the total energy E reachesits minimum of Emin ¼ Cv�2dc=4, with v�dc is the desired value of DC

0.4 0.5 0.6 0.7 0.8

Time (s)

0.4 0.5 0.6 0.7 0.8

Time (s)

0.4 0.5 0.6 0.7 0.8

Time (s)

tinued)

642 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

voltage [43]. This condition is called the minimum energy propertywhich can be used as the basic principle for DC-capacitor voltagesbalancing and control [45]. The adopted control method shouldminimize the quadratic cost function J associated with voltage devi-ation of the DC-capacitors. The cost function is defined as follows:

J ¼ C2ðDv2

C1 þ Dv2C2Þ ð33Þ

where

DvCj ¼ vCj � v�dc=2; j ¼ 1;2

The mathematical condition to minimize J is:

dJdt¼ DvC1iC1 þ DvC2iC2 6 0 ð34Þ

where iCj (j = 1,2) is the current through capacitor Cj. These currentsare affected by the DC-side intermediate branch currents, i0 and i1.

0 0.05 0.1 0.15-400

-200

0

200

400

T

Sour

ce v

olta

ges

(V)(a)

0 0.05 0.1 0.15

-50

0

50

T

Sour

ce c

urre

nts

i sab

c (A

)

(b)

0 0.05 0.1 0.15

-50

0

50

T

Loa

d cu

rren

ts i

Lab

c (A

)(c)

(d)

0 0.05 0.1 0.15

-40

-20

0

20

40

T

(d)iFa

iFb

iFc

iFn

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-15

-10

-5

0

5

10

15

Time (s)

Neu

tral

cur

rent

isn

(A

)

(e)

cur

rent

s i F

abcn

(A

)SA

PF

Fig. 13. Simulation results of the proposed DP/CC-3DSVM us

These currents can be calculated if the switching states used in theswitching pattern are known.

Thus, it is advantageous to express (34) in terms of i0, and i1. TheDC-capacitor currents are expressed as:

iC1 ¼ i1

iC2 ¼ i1 þ i0ð35Þ

By substituting iC2 and iC1 given by (35) in (34), the condition toachieve voltage balancing is deduced as:

DvC1i1 þ DvC2ði1 þ i0Þ 6 0 ð36Þ

When the DC link voltages vC1 and vC2 are close to their referencev�dc=2, the following condition is verified :

DvC1 þ DvC2 ¼ 0 ð37Þ

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

1

2

3

x 104

Time (s)

(f)

p s(W

) an

d q

s(V

ar)

Sour

ce p

ower

s

ing nonlinear controller under unbalanced grid voltage.

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 643

Using (37), Eq. (36) can be written as:

DvC2i0 6 0 ð38Þ

Applying the averaging operator, over one sampling period, to(38) results in:

1Ts

Z ðkþ1ÞTs

kTs

ðDvC2i0Þdt 6 0 ð39Þ

Assuming that sampling period Ts, is adequately small as com-pared to the time interval associated with the dynamics of capac-itor voltages. These letter can be assumed to remain constant overone sampling period [45] and (39) is consequently simplified to:

DvC2ðkÞ�i0 6 0 ð40Þ

where DvC2(k) is the voltage drift at sampling period k, and �i0 is theaveraged value of the i0.

0 0.05 0.1 0.15-400

-200

0

200

400

T

Sour

ce v

olta

ges

(V)(a)

0 0.05 0.1 0.15

-50

0

50

T

Sour

ce c

urre

nts

i sab

c (A

)

(b)

0 0.05 0.1 0.15

-50

0

50

T

Loa

d cu

rren

ts i

Lab

c (A

)(c)

0 0.05 0.1 0.15-50

0

50

T

SAP

F c

ourr

ents

iF

abcn

(A

)(d) iFa

iFb

iFc

iFn

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-15

-10

-5

0

5

10

15

Time (s)

Neu

tral

cur

rent

in (

A)

(e) (

Fig. 14. Simulation results of the proposed DP/CC-3DSVM

The relationship between the DC-intermediate branch current i0and AC-currents (iFa, iFb and iFc) for different switching states isrequired.

The current�i0 should be computed for different combinations ofadjacent redundant switching states over a sampling period andthe best combination which minimizes (40) is selected.

The implementation procedure of the 3DSVM with proposedalgorithm and voltage balancing strategy is summarized in Fig. 7.

Simulation results

For verifying the validity of the proposed control of the three-level four-leg SAPF; and due to lack sufficient hardware in our lab-oratory to implement this complexe control, computer simulationsunder different load and source voltage conditions are carried outon the MATLAB/Simulink. The simulation parameters are shown inTable 4.

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0.2 0.25 0.3 0.35

ime (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

1

2

3

x 10 4

Time (s)

Sour

ce p

ower

s p s(W

) an

d q s

(Var

)f)

using PI controller under unbalanced grid voltage.

Fig. 15. Frequency spectrum of source current, (a) using nonlinear controller, (b) using PI controller.

2 4 6 8 100

2

4

6

8

10

Switching frequency f s (kHz)T

HD

(%

)

PI controller

Nonlinear controller

Fig. 16. Source current THD versus value of switching frequency.

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.2

0.4

0.6

0.8

1

1.2

DC reference voltage (kV)

DC

cap

acit

ors

volt

ages

rip

ples

(%

)

Fig. 17. DC capacitors voltages ripple versus value of DC voltage.

644 M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646

The source current of the first phase and its harmonic spectrumbefore and after compensation are illustrated in Figs. 8–10. Itresults that the SAPF decreases the total harmonic distortion(THD) in the source currents from 17.08% to 2.77% with PI control-ler. However, with nonlinear controller, the THD is reduced to0.22% which proves the effectiveness of the proposed nonlinearcontroller.

Figs. 11 and 12 present the dynamic behavior of the system fornonlinear and PI controller respectively, where the four-leg SAPF isswitched on 0.2 s later. After 0.5 s, an additional load is added insingle-phase diode bridge rectifier in phase (b) in order to inducean unbalanced load.

It can be observed that the three-phase source currents are bal-anced and sinusoidal after compensation in two control methodsfor this case. As shown in Figs. 11d and 12d, the neutral currentis almost canceled with a low ripple in case where the nonlinearcontrol is applied (2.86% for the nonlinear controller and 18.57%for the PI controller).

In Figs. 11e and 12e, one can see that the active power joined itsnominal value and that the reactive power becomes null when theactive filter is activated at time 0.2 s. For clarity, a phase-a of sourcecurrent and its corresponding phase voltage are shown for illustra-tion in Figs. 11h and 12h. It can be observed that the unity powerfactor operation is successfully achieved.

The DC bus voltage variation due to the load change is about30 V, and the recovery time of DC voltage is about 0.1 s. The capac-itor voltages on the DC bus are balanced before and after the loadvariation. However, the absence of overshoots in DC voltagesresponses during nonlinear load change, and low neutral currentand power ripples, demonstrates the superiority of the nonlinearcontroller compared to its counterpart traditional PI controller.

The simulation results with the nonlinear and PI controllerunder unbalanced grid voltages are shown in Figs. 13 and 14respectively. The neutral current is eliminated and the reactivepower compensation is successfully achieved using nonlinear con-trol method. As shown in Fig. 15 the total harmonic distortion insource currents is 3.58% with PI control and 2.85% with nonlinearcontrol. So, the distortion in source current with nonlinear controlunder unbalanced grid voltage is also less than in case of PI controlmethod.

From the Fig. 16, one notices well that the THD decreasesremarkably with the increase of the switching frequency. It canbe concluded that the DP/CC-3DSVM based on nonlinear controllercan operate with reduced switching frequency (1–2 kHz).

According to Fig. 17, the ripple of DC capacitors voltagesdecreases with the increase in the DC voltage value vdc, which leadsconsequently to decrease the modulation index ðm ¼

ffiffiffi2p

U�Fab=vdcÞ.Indeed, with a smaller m, the number of redundant switchingstates increases, which provides a higher degree of freedom to bal-ance the DC capacitors voltages, and consequently, the ripple ofthese voltages becomes small.

5. Conclusion

This work presented a theoretical study with simulation ofhybrid direct power/current control combined with feedback line-arization control for a three-level four-leg shunt active power filter.The performances of the active power filtering system based onnonlinear controller are analyzed and compared with conventionalcontroller for different disturbed operating conditions. The pro-posed control scheme gives high performance under both dynamicand steady state operations in terms of the current harmonics fil-tering, reactive power compensation, source current balancing,and neutral current elimination. In this paper three-level 3DSVMwith a new algorithm is also presented for three-level four-leg con-verter. The significant outcome of the proposed algorithm is itsinherent simplicity. Unlike the conventional 3DSVM algorithm thatrequires reference vector determination, on-duration time calcula-tion and pulses creation in all sectors. Therefore, the proposed3DSVM algorithm is much simpler and easier for digital implemen-tation since it reduces the hardware and software complexity anddecreases the required computational time. Furthermore, the cor-rection of the inherent instability of the DC link capacitor voltageshas been considered. Indeed, the objective of maintaining balancedvoltages in DC-link capacitors is carried out effectively with theadopted three-level 3DSVM.

M. Bouzidi et al. / Electrical Power and Energy Systems 61 (2014) 629–646 645

Appendix

Table: Interchanging the switching states of each tetrahedronlocated in same prism for sector one and two.

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