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IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 12, NO. 3, JUNE 2016 1035

Modeling, Digital Control, and Implementation ofa Three-Phase Four-Wire Power Converter

Used as a Power Redistribution DeviceAniel Silva de Morais, Fernando Lessa Tofoli, and Ivo Barbi, Fellow, IEEE

Abstract—This paper presents a power electronicconverter used to redistribute the power among the phasesin unbalanced power systems, which is supposed to bedesigned based on the involved degree of unbalance. Abidirectional converter is chosen for this purpose, whosemodeling is presented in the dq0 system. This solution canbe considered as part of a unified control system, whereconventional active power filters may be solely respon-sible for compensating harmonics and/or the tuning ofpassive filters becomes easier, with consequent reductionin involved costs in a decentralized approach. The adoptedcontrol strategy is implemented in digital signal processorTMS320F2812, while experimental results obtained froman experimental prototype rated at 17.86 kVA are properlydiscussed considering that the converter is placed at thesecondary side of a transformer supplying three distinctsingle-phase loads. It is effectively shown that the con-verter is able to balance the currents in the transformerphases, thus leading to the suppression of the neutralcurrent.

Index Terms—Active power filters (APFs), harmonics,power redistribution, reactive power compensation.

I. INTRODUCTION

D ISTRIBUTION power systems are intrinsically dynami-cal considering that the demand is not constant over time

and the phase loading is typically asymmetrical. Such importantissues make the expansion planning essential for the accurateoperation of systems in face of continuous changes in demandand also equipment utilization factor [1].

The operation under balanced conditions is better recom-mended so that the utilization factor of transformers and feederscan be improved. Thus, it is possible to minimize the risk of

Manuscript received October 09, 2014; revised October 01, 2015 andJanuary 12, 2016; accepted December 22, 2015. Date of publicationMarch 20, 2016; date of current version June 02, 2016. This work wassupported in part by Conselho Nacional de Desenvolvimento Científicoe Tecnológico (CNPq), in part by Coordenação de Aperfeiçoamento dePessoal de Nível Superior (CAPES), in part by Fundação de Amparo àPesquisa do Estado de Minas Gerais (FAPEMIG), and in part by InstitutoNacional de Energia Elétrica (INERGE). Paper no. TII-15-1117.

A. S. deMorais is with the Federal University of Uberlândia,Uberlândia 38408-100, Brazil (e-mail: [email protected]).

F. L. Tofoli is with the Universidade Federal de São João del-Rei, SãoJoão del-Rei 36307-352, Brazil (e-mail: [email protected]).

I. Barbi is with the Federal University of Santa Catarina, Florianópolis88040-970, Brazil (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TII.2016.2544248

overload associated to several phenomena such as the fluctua-tion of line voltage. It is also important to mention that load bal-ancing can be achieved through the reconfiguration of distribu-tion networks by using switches or rearranging the load amongthe system phases [2]. Besides the reconfiguration of networks,the use of alternative solutions is a must when the adoptionof such classical approaches becomes unfeasible while loadbalancing is not possible over the entire operation period [3].

The presence of reactive power and harmonics in power sys-tems has become a continuous matter of studies for engineersand researchers by the development of reactive compensatorsand filters [4]. Typically, reactive power can be compensatedby using passive elements such as inductors and capacitors ina solution that aggregates increased robustness and appreciablesize, weight, and volume. Besides, there is a risk of resonancedue to power system harmonics. On the other hand, harmonicscan be compensated with tuned passive filters (PFs) due to theinherent simplicity and satisfactory frequency response charac-teristics that can be obtained, also considering the reduced cost[5]–[8]. However, the aforementioned solution presents somepractical limitations considering that the grid impedance is typ-ically unknown and may affect the filter performance, while itmay not respond adequately as well if the load power factorcomes to vary, for instance.

The use of static power converters in power systems becamewidespread with the development of three-phase active powerfilters (APFs). Even though the basic premises were intro-duced during the 1970s, it can be stated that the conceptionof the instantaneous power theory in [9] is responsible fortheir increasing popularity because it allows the real-timecompensation.

Considering the application of power electronic devices todistribution systems, this paper presents the power balancingconcept using the power redistribution in three-phase four-wiresystems. For this purpose, a static power converter is placedat the secondary side of a distribution transformer, so that itcan be seen as a balanced three-phase three-wire system. Theequipment is the so-called power redistribution device (PRD),as the involved ratings are chosen based on the degree of thephase unbalance, while a detailed design procedure is presentedas one of the main contributions.

Conventional shunt APFs are able to balance the phasecurrents and also compensate reactive power and harmonics.However, they are often designed for the worst case scenarioinvolving linear and nonlinear loads as based on experimentalmeasurements in order to define the involved ratings using

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1036 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 12, NO. 3, JUNE 2016

typically oversized elements. Since all aforementioned tasksare addressed to a single device, wide passing band and fastdynamic response are a must, while appreciable cost existsin this case. Besides, the switching frequency of the powerelectronic converter must be high, with consequent increase inswitching losses. The PRD is primarily designed to be placedat the secondary side of a distribution transformer in order tobalance the phase currents. This task involves low-frequencycurrents, while the PRD is designed with narrower passingband than the traditional shunt APF and reduced cost. In thiscase, lower switching frequencies can also be used in the powerconverter as higher efficiency is expected.

A conceptual compensation system can be composed of pas-sive elements to compensate reactive power, the PRD to balancethe system currents, and an APF responsible for compensat-ing harmonics, considering that all devices perform individualand independent tasks, being controlled by a unified supervi-sory system in a flexible and decentralized approach. If the APFis solely responsible for compensating harmonics, the involvedratings and cost are supposed to be further reduced. Of course,there is still the cost associated to the PRD, although it aims atbalancing the system currents only. It is worth to mention thatthe PRD is not supposed to be designed to perform all tasks,but the experimental results presented in this paper show that itmay act as a conventional shunt APF.

A methodology for the analysis, control, and application ofthe PRD is discussed. A proper control strategy adequate toachieve balance of power system currents or powers is also pro-posed based on the Park and Clark transforms. Some distinctadvantages of the proposal are the postponing of investments inthe distribution system; reduction in losses due to elimination ofneutral currents and voltage drops across the secondary wind-ings of transformers; increase in utilization factor of relatedequipment; and improvement in reliability and power qualityindices.

II. POWER REDISTRIBUTION CONCEPT

Considering a typically unbalanced three-phase four-wiresystem, e.g., the distribution network, harmonics, and reactivepower flow exist since nonlinear and/or asymmetrical loads aresupplied. Budeanu’s concept of reactive and distortion powerhas been widely used to quantify the presence of nonlinearloads in power circuits, even though it has been heavily criti-cized [10]. This is the main reason why novel definitions forsuch quantities have been proposed in literature [11].

As it was mentioned before, conventional APFs are normallydesigned to provide reactive power and harmonic compensa-tion simultaneously [12]–[15]. It is worth to mention that theactive power redistribution is also possible considering that agiven phase with highest loading can send power to the remain-ing ones so that balanced currents exist, while only positivesequence components are supposed to flow through the system.

By using the scheme proposed in this work, it is possibleto compensate reactive power and harmonics, as well as bal-ancing the currents through the system phases. The converteraims at arranging the power among the phases through the bal-ance of the active and reactive components of the instantaneousthree-phase power. The redistribution of reactive power can be

achieved by using single-phase passive elements to absorb orsupply the necessary amount of power that allows balancingthe phases. However, the redistribution of active power is onlypossible by considering some sort of “coupling” among thebranches.

Let us consider a hypothetical example, where the secondaryside of a three-phase transformer supplies an unbalanced three-phase three-wire load. In this case, a PRD is responsible forcompensating the resulting unbalance. For this purpose, onemust determine the total apparent power (which is not theinstantaneous power), the desired input power for each phase,and the supplementary power which is supposed to be absorbedor supplied by the PRD. The total load apparent power ST isobtained by summing the apparent powers in each load branch,i.e., Sa, Sb, and Sc

ST = Sa + Sb + Sc =∣∣∣−→S T

∣∣∣ . (1)

Besides, the total load apparent power can be divided into anactive component P and a reactive component Q as

−→S T = P + jQ. (2)

In order to keep balanced power among the phases, thefollowing statement must be obeyed:

−→S T1 =

−→S T2 =

−→S T3 =

P + jQ

3= P ′ + jQ′ (3)

where−→S T1,

−→S T2, and

−→S T3 are the input apparent powers; and

P ′ and Q′ are their respective active and reactive components.The powers processed by the PRD in each one of the phases

are S1, S2, and S3, as the following assumption is valid:

−→S 1 +

−→S 2 +

−→S 3 = 0. (4)

In order to operate as a PRD, the converter must not pro-cess the average three-phase instantaneous power (except forthe converter losses that exist in the form of active power) andthe average imaginary power. In other words, the grid powerthat enters the power electronic converter is null if power lossesare neglected.

The grid power for each phase can be obtained by summingthe power processed by the PRD and the load power as⎧⎪⎪⎨

⎪⎪⎩−→S T1 =

−→S 1 +

−→S a

−→S T2 =

−→S 2 +

−→S b

−→S T3 =

−→S 3 +

−→S c

⎫⎪⎪⎬⎪⎪⎭ . (5)

III. CONCEPTION OF THE PRD

The power redistribution concept was analyzed in the pre-vious section. Considering that three wye-connected currentsources are able to rearrange the power in an unbalanced powersystem, it is necessary to use a static converter with such charac-teristic. The three-phase four-wire bidirectional boost convertershown in Fig. 1 is used for this purpose, where i1(t), i2(t),and i3(t) are the instantaneous currents through the converter;io1(t) and io2(t) are the instantaneous output currents; vo1(t)

DE MORAIS et al.: MODELING, DIGITAL CONTROL, AND IMPLEMENTATION OF THREE-PHASE FOUR-WIRE POWER CONVERTER 1037

Fig. 1. Half-bridge bidirectional boost converter used as a PRD.

and vo2(t) are the instantaneous output voltages; iN (t) is theinstantaneous neutral current; L1 = L2 = L3 are the input fil-ter inductors; and Rs models the losses associated to passiveelements and semiconductors that exist in the converter givenby PLoss. Of course, this converter allows bidirectional energyflow between the dc link and the ac input voltage sources.

A. Converter Modeling

The converter modeling can be performed by using the Parkand Clark transforms in the dq0 system, while the study con-sists in determining not only the transfer functions of the inputcurrents to the duty cycles, but also the transfer functions ofthe output voltage to the input currents. The respective small-signal model can be easily derived according to the guidelinesprovided in [16].

Let us considerer that vX(t), vY (t), and vZ(t) correspondto voltage sources which depend on given switching statesaccording to the following pattern:{S1→1S4→0

⇒ vX (t)=vo1 (t)

} {S1→0S4→1

⇒ vX (t)=− vo2 (t)

}{S2→1S5→0

⇒ vY (t)=vo1 (t)

} {S2→0S5→1

⇒ vY (t)=− vo2 (t)

}{S3→1S6→0

⇒ vZ (t)=vo1 (t)

} {S3→0S6→1

⇒ vZ (t)=− vo2 (t)

}.

(6)

According to (6), it is possible to define vX(t), vY (t), andvZ(t) over one switching cycle as a function of duty cyclesd1(t), d2(t), and d3(t), i.e.,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

vX(t) = d1(t) · vo1 (t) + [d1(t)− 1] · vo2 (t)= d1(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

vY (t) = d2(t) · vo1 (t) + [d2(t)− 1] · vo2 (t)= d2(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

vZ(t) = d3(t) · vo1 (t) + [d3(t)− 1] · vo2 (t)= d3(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (7)

Considering that the three-phase converter can be repre-sented as three single-phase systems, the following expressionresults:⎧⎨

⎩v1(t) = L · d

dt [i1(t)] +Rs · i1(t) + vX(t)v2(t) = L · d

dt [i2(t)] +Rs · i2(t) + vY (t)v3(t) = L · d

dt [i3(t)] +Rs · i3(t) + vZ(t)

⎫⎬⎭ . (8)

Substituting (7) in (8) gives⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

v1(t) = L · d

dt[i1(t)] +Rs · i1(t)

+d1(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

v2(t) = L · d

dt[i2(t)] +Rs · i2(t)

+d2(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

v3(t) = L · d

dt[i3(t)] +Rs · i3(t)

+d3(t) · [vo1 (t) + vo2 (t)]− vo2 (t)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (9)

Let us define vectors−→V 123,

−→I 123,

−→d 123,

−→d 123,

−→V 0dq ,−→

I odq , and−→d odq as

−→V 123 =

⎡⎣ v1(t)v2(t)v3(t)

⎤⎦−→

I 123 =

⎡⎣ i1(t)i2(t)i3(t)

⎤⎦−→

d 123 =

⎡⎣ d1(t)d2(t)d3(t)

⎤⎦

−→d 123 =

⎡⎣ d1(t)− 1d2(t)− 1d3(t)− 1

⎤⎦−→V 0dq =

⎡⎣ v0(t)vd(t)vq(t)

⎤⎦

−→I odq =

⎡⎣ i0(t)id(t)iq(t)

⎤⎦−→

d odq =

⎡⎣ d0(t)dd(t)dq(t)

⎤⎦ . (10)

By manipulating expression (9), the input currents can beobtained in the vector form, i.e.,

−→I 123

−→I 123=

1

L

∫ {−→V 123−−→

d 123 · [vo1 (t)+vo2 (t)] +vo2 (t)−Rs·−→I 123

}.

(11)

The output currents can be calculated from Fig. 1, whichallows obtaining the currents through the output capacitors as{

io1(t)=i1(t)· d1 (t)+i2 (t)· d2 (t)+i3 (t)· d3 (t)io2 (t)=i1(t)· d1(t)+i2(t)· d2(t)+i3(t)· d3(t)−iN (t)

}

(12)

where the neutral current is iN (t) = io1 (t)− io2 (t).From expressions (10) and (12), the output voltages can be

calculated as⎧⎪⎪⎨⎪⎪⎩

vo1 (t)=1

C

∫ (−→I 123

T · −→d 123

)

vo2 (t)=1

C

∫ (−→I 123

T ·−→d 123

)=vo1 (t)− 1

C

∫iN (t)

⎫⎪⎪⎬⎪⎪⎭ .

(13)

The transfer functions of the input currents i0(s), id(s), andiq(s) to the respective duty cycles d̃0(s), d̃′d(s), and d̃′q(s) canthen be written as⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

i0(s)

d̃0(s)= − Vo

L · s+RS

id(s)

d̃′d(s)= − Vo

L · s+RSiq(s)

d̃′q(s)

= − Vo

L·s+RS

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(14)


where d̃0(s), d̃d(s), and d̃q(s) are the duty cycles referred tothe small-signal model of the converter.

Besides, the transfer functions of the output voltage tocurrents id(s), iq(s), and i0(s) are⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

vo (s)

id (s)=

√6 · Vpk

s · C · Vo−√

8

3· Sprocess

s · C · Vo · Vpk· ρ · (L · s+2 ·Rs)

vo (s)

iq (s)= −

√8

3· Sprocess

s · C · Vo · Vpk· (L · s+ 2 ·Rs)

vo (s)

i0 (s)= 0

vdiff (s)

i0 (s)=

√3

s · C

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(15)

where vdiff(s) corresponds to the difference between the volt-ages across the dc links, Vpk is the peak value of the phase inputvoltage, Sprocess is the apparent power processed by the con-verter, L is the inductance of the input filter inductors, C isthe capacitance of the output filter capacitors, and ρ is a factorthat relates the losses and the processed apparent power, beingarbitrarily chosen in the design procedure.

B. Power Unbalance Standard Deviation

Balancing the total complex power is based on balancedloads connected to each phase of the three-phase distributiontransformer. According to [17], the complex power standarddeviation ΔS can be expressed as

ΔS =

√1

3

∑p=a,b,c

(S0 − Sp)2 (16)

S0 =Sa + Sb + Sc

3(17)

where S0 and Sp represent the mean apparent power and theapparent power in a given phase p, respectively. If ΔS = 0, thetransformer is perfectly balanced.

It is reasonable to state that ΔS represents the standard devia-tion regarding the apparent power in phases a, b, and c. Besides,the dispersion around the mean apparent power for each phaseis defined as ΔSa,ΔSb, and ΔSc

ΔSa = |S0 − Sa| (18)

ΔSb = |S0 − Sb| (19)

ΔSc = |S0 − Sc| . (20)

Therefore, expression (16) can be rewritten as

ΔS =

√1

3· (ΔSa

2 +ΔSb2 +ΔSc

2). (21)

Let us consider two possible cases where unbalance exists ina three-phase transformer. The best scenario occurs if Sa1 =12 kVA, Sb1 = 8 kVA, and Sc1 = 10 kVA, as S0 = 10 kVAaccording to (17). Besides, ΔSa1 = 2 kVA, ΔSb1 = 2 kVA,and ΔSc1 = 0 can be calculated using expressions (18)–(20).

Therefore, expression (22) becomes valid⎧⎪⎨⎪⎩

ΔSa1 = ΔSb1 +ΔSc1

ΔSb1 = ΔSa1

ΔSc1 = 0

⎫⎪⎬⎪⎭ . (22)

Substituting (22) in (21) gives

ΔSa1∼= 1.225 ·ΔS1. (23)

The worst scenario occurs if Sa2 = 12 kVA, Sb2 =9 kVA, and Sc2 = 9 kVA, and, consequently, S02 = 10 kVA.Analogously, ΔSa2 = 2 kVA, ΔSb1 = 1 kVA, and ΔSc1 =1 kVA are obtained. Besides, expression (24) can be written⎧⎨

⎩ΔSa2 = ΔSb2 +ΔSc2

ΔSb2 = ΔSc2 =ΔSa2

2

⎫⎬⎭ . (24)

Substituting (24) in (21) gives

ΔSa2∼= 1.414 ·ΔS2. (25)

From (23) and (25), it is reasonable to state that it is possi-ble to determine the operating range for the phase that presentshighest dispersion of the apparent power, which is 1.225 ·ΔS ≤ ΔSa ≤ 1.414 ·ΔS . In other terms, it is possible to esti-mate the converter rated apparent power SPRD considering theworst case ΔSa = 1.414 ·ΔS , resulting in SPRD = 3 ·ΔSa =4.24 ·ΔS in this case.

If the standard deviation is supposed to be expressed as apercent quantity ΔS%

, the following expression results:

ΔS(%) =ΔS

S0· 100% =

3 ·ΔS

S3φ· 100% (26)

where

S3φ = Sa + Sb + Sc. (27)

Besides, the following percent parameter can be defined forthe converter rated power:

SPRD(%) =SPRD

S3φ· 100% =

4.24 ·ΔS

S3φ· 100%. (28)

The ratio between SPRD(%) and ΔS%can be obtained from

(26) and (28) as

SPRD(%) =(4.24 ·ΔS)/S3φ

(3 ·ΔS)/S3φ·ΔS(%) = 1.414 ·ΔS(%). (29)

The amount of power that is effectively processed by thePRD can be defined as

Sprocess = ΔSa +ΔSb +ΔSc. (30)

For the worst case described according to (24), expression(31) results

Sprocess = 2 ·ΔSa = 2.828 ·ΔS (31)

Sprocess

SPRD=

2 ·ΔSa

3 ·ΔSa=

2

3. (32)

It is important to notice that the maximum apparent powerprocessed by the converter is equal to two thirds of the ratedpower. Besides, the following expression is also valid:

Sprocess(%) =2

3· SPRD(%) = 0.9428 ·ΔS(%). (33)


C. Converter Design

The three-phase bidirectional converter shown in Fig. 1 canbe designed for power redistribution purposes. Due to irregularand unpredictable operating conditions, it is important to men-tion that there is not a rated situation, but a maximum limit thatmust be respected. In order to design the converter, the maxi-mum processed complex power is considered, where each leg issupposed to process up to one third of the aforementioned limit.

1) Preliminary Calculation: The modulation index can beobtained as

m =2 · Vpk

Vo. (34)

Let us consider that the converter rated power is SPRD =3 ·ΔSa = 4.24 ·ΔS as stated in Section III-B. Then, the rmsinput current is given by

Irms =SPRD

3 · Vrms(35)

where Vrms is the RMS value of the phase input voltage.The peak input current, which does not consider the ripple

current, is obtained as

Ipk =√2 · Irms. (36)

The input current ripple is defined as

ΔIL =ΔIL(%)

100· Ipk. (37)

The output voltage ripple and the converter losses can becalculated from the following expressions:

Δvo =Δvo(%)

100· Vo

2(38)

Rs =ρ · SPRD

3 · I2rms. (39)

2) Semiconductor Elements: When operating as a recti-fier, energy flows from the input voltage source to the outputcapacitors. When operating as an inverter, the opposite occurs.In the latter case, the current is phase-shifted by 180◦ withrespect to the voltage. From the analysis of the operating stagesof the converter shown in Fig. 1, it is possible to define the cur-rent and voltage stresses regarding the semiconductor elements.Considering the analogy between the operation stages for bothmodes, it is possible to determine the expressions that allow theproper design of the semiconductor elements.

The peak, average, and rms values of the current through boththe main switches and diodes are given by expressions (40),(41), and (42), respectively

IS(pk) = ID(pk) = Ipk +ΔIL2

(40)

IS(avg) = ID(avg) =Ipk

2 · π ·(1 +

π ·m4

)(41)

IS(rms) = ID(rms) = Ipk ·√

1

8+

m

3 · π . (42)

The maximum voltage across the main switches and thediodes is

VS(max) = VD(max) = Vo +Δvo2

. (43)

Fig. 2. Experimental prototype.

TABLE IDESIGN SPECIFICATIONS

3) Output Capacitors: The rms current trough one of theoutput capacitors is

Io1(rms) =

√2

3· Ipk. (44)

Besides, the capacitance can be determined as

Co1 = Co2 = C =Ipk

π · fL ·Δvo(45)

where fL is the grid frequency.4) Boost Inductor: The peak and rms currents through the

boost inductor are given by (46) and (47), respectively

IL(rms) = Irms (46)

IL(pk) = Ipk +ΔIL2

. (47)

The boost inductor can be determined as

L =0.259 · Vo

fs ·ΔIL(48)

where fs is the switching frequency.


TABLE IIMEASURED RESULTS

Fig. 3. Load currents and neutral current.

IV. EXPERIMENTAL RESULTS

In order to validate the theoretical assumptions, the exper-imental prototype shown in Fig. 2 was designed according tothe parameters given in Table I. The control strategy was imple-mented in digital signal processor (DSP) TMS320LF2812.Three single-phase loads are chosen to demonstrate the perfor-mance of the PRD, which is not only able to provide and/orabsorb active and or/reactive power, but also compensate har-monics. As it was mentioned before, the PRD can be alsoassociated to a typical APF solely responsible for compensatingharmonics in a more flexible compensation approach.

1) Phase 1: Linear load composed by a series RL (resistor-inductor) load, where R = 0.38 Ω and L = 22.5 mH.

2) Phase 2: Nonlinear load composed by a single-phasediode rectifier with input impedance Li = 430 µH andfilter capacitor Cf = 9.4 mF, supplying a resistive loadRo = 37 Ω.

3) Phase 3: Linear load composed by a series RL load,where R = 4.5 Ω and L = 1 mH.

Fig. 4. Currents supplied by the PRD.

The converter was thoroughly evaluated and the resultsregarding each one of the phases are presented in Table II interms of the active power P , reactive power Q, and distortedpower D as calculated by the oscilloscope software. The totalharmonic distortion (THD) of the current in phase 2 is muchhigher than those in phase 1 and 3 due to the presence of anonlinear load. In this case, the converter is responsible for pro-cessing 11.73 kVA to redistribute the power among the systemphases.

The supply voltages are nearly balanced according toTable II. Fig. 3 presents the phase currents in the aforemen-tioned three-phase load. It can be seen that currents Ia andIc are sinusoidal and represent an RL load, while current Ibcorresponds to a nonlinear load whose harmonic content is typ-ically high. Besides, the neutral current is appreciable, whosewaveform clearly shows that the load is noticeably unbalanced.

In this case, the PRD is supposed to rearrange the poweramong the phases so that the system becomes perfectly bal-anced, i.e., the phase currents in the transformer are expected tobe sinusoidal and phase-shifted by 120◦. In other words, poweris taken from phase c, which is heavily loaded according to


Fig. 5. Currents in the secondary side of the transformer.

Fig. 6. Transformer neutral current and phase voltage v3(t).

Fig. 7. Detailed view of the converter current during a sudden positivestep from no-load to rated load condition.

Table II, and sent to phases a and b. For this purpose, the PRDis responsible for supplying the currents shown in Fig. 4. Onceagain, it can be seen that current IB represents a nonlinear load.

Fig. 5 shows the transformer phase currents, while the neu-tral current is presented in Fig. 6. The current in phase 2 of the

transformer presents some notching due to the high harmoniccontent of the load current (i.e., diode rectifier), as the con-verter is not able to mitigate its influence completely. However,it can be seen that the transformer neutral current is practicallyeliminated as desired.

Finally, Fig. 7 shows a positive load step, thus demonstratingthe accurate performance of the control system.

V. CONCLUSION

The design and implementation of a digitally controlled four-wire converter to balance the power among the phases of adistribution transformer have been presented in this paper. Aprototype rated at 17.86 kVA has been implemented, whichprocesses 11.73 kVA to redistribute the complex powers thatinvolve three distinct loads. It can be seen that the converter isable to provide the balance of phase currents and mitigation ofneutral current in the transformer.

The power redistribution concept has been defined as a sim-ple strategy that allows estimating the required amount of powerfor the design of the converter that must be used. In order torearrange the complex power among the system phases usingthe proposed PRD, the rms phase currents must be nearly thesame, as the utilization factor of equipment such as cables andfeeders is consequently improved. The design procedure of thestudied converter is based on the involved degree of unbalanceas shown in the theoretical analysis, which can be stated as oneof the main contributions of this work.

Considering that the system currents are balanced by thePRD, which is supposed to perform a single task in a conceptualglobal compensation systems, passive elements can be used toprovide or absorb reactive power, with reduced cost if comparedwith the use of traditional APFs. On the other hand, APFs canbe solely designed to compensate harmonics, while their associ-ated cost is minimized by using this approach because the ratedpower processed by the equipment is reduced. In this case, theuse of hybrid filters can also be considered in other to furtherreduce cost. It is also important to mention that if the poweris not balanced, PFs do not represent a viable solution sincethe reactive power flowing through each phase is not the same.The aforementioned simultaneous approaches can be integratedand properly managed by a unified control system in a flexiblesolution.

REFERENCES

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[2] G. Buticchi, L. Consolini, and E. Lorenzani, “Active filter for the removalof the dc current component for single-phase power lines,” IEEE Trans.Ind. Electron., vol. 60, no. 10, pp. 4403–4414, Oct. 2013.

[3] V. F. Corasaniti, M. B. Barbieri, and P. L. Arnera, “Compensation withhybrid active power filter in an industrial plant,” IEEE Latin Amer. Trans.,vol. 11, no. 1, pp. 447–452, Feb. 2013.

[4] T.-H. Chen and J.-T. Cherng, “Optimal phase arrangement of distribu-tion transformers connected to a primary feeder for system unbalanceimprovement and loss reduction using a genetic algorithm,” IEEE Trans.Power Syst., vol. 15, no. 3, pp. 994–1000, Aug. 2000.


[5] D. Lamar, J. Sebastian, M. Arias, and A. Fernandez, “On the limit ofthe output capacitor reduction in power-factor correctors by distortingthe line input current,” IEEE Trans. Power Electron., vol. 27, no. 3,pp. 1168–1176, Mar. 2012.

[6] A. M. Al-Zamil and D. A. Torrey, “A passive series, active shunt filter forhigh power applications,” IEEE Trans. Power Electron., vol. 16, no. 1,pp. 101–109, Jan. 2001.

[7] X. Wang, F. Blaabjerg, and P. C. Loh, “Virtual RC damping of LCL-filtered voltage source converters with extended selective harmonic com-pensation,” IEEE Trans. Power Electron., vol. 30, no. 9, pp. 4726–4737,Sep. 2015.

[8] A. F. Zobaa, “Optimal multiobjective design of hybrid active power filtersconsidering a distorted environment,” IEEE Trans. Ind. Electron., vol. 61,no. 1, pp. 107–114, Jan. 2014.

[9] H. Akagi, Y. Kanazawa, and A. Nabae, “Instantaneous reactive powercompensators comprising switching devices without energy storage com-ponents,” IEEE Trans. Ind. Appl., vol. IA-20, no. 3, pp. 625–630,Apr./May 1984.

[10] L. S. Czarnecki, “What is wrong with the Budeanu concept of reactive anddistortion power and why it should be abandoned,” IEEE Trans. Instrum.Meas., vol. IM-36, no. 3, pp. 834–837, Sep. 1987.

[11] L. S. Czarnecki, “Distortion power in systems with nonsinusoidal volt-age,” Proc. Inst. Elect. Eng. B Elect. Power Appl., vol. 139, no. 3,pp. 276–280, May 1992.

[12] V. Khadkikar, A. Chandra, and B. Singh, “Digital signal processorimplementation and performance evaluation of split capacitor, four-legand three h-bridge-based three-phase four-wire shunt active filters,” IETPower Electron., vol. 4, no. 4, pp. 463–470, Apr. 2011.

[13] S. Rahmani, N. Mendalek, and K. Al-Haddad, “Experimental design ofa nonlinear control technique for three-phase shunt active power filter,”IEEE Trans. Ind. Electron., vol. 57, no. 10, pp. 3364–3375, Oct. 2010.

[14] A. Bhattacharya and C. Chakraborty, “A shunt active power filter withenhanced performance using ANN-based predictive and adaptive con-trollers,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 421–428, Feb.2011.

[15] M. Popescu, A. Bitoleanu, and V. Suru, “A DSP-based implementationof the p–q theory in active power filtering under nonideal voltage con-ditions,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 980–989, May2013.

[16] A. S. Morais and I. Barbi, “A control strategy for four-wire shunt activefilters using instantaneous active and reactive current method,” in Proc.32nd IEEE Annu. Conf. Ind. Electron., 2006, pp. 1846–1851.

[17] T.-H. Chen and J.-T. Cherng, “Optimal phase arrangement of distribu-tion transformers connected to a primary feeder for system unbalanceimprovement and loss reduction using a genetic algorithm,” IEEE Trans.Power Syst., vol. 15, no. 3, pp. 994–1000, Aug. 2000.

Aniel Silva de Morais received the B.Sc. andM.Sc. degrees in electrical engineering from theFederal University of Uberlândia, Uberlândia,Brazil, in 2002 and 2004, respectively, andthe Ph.D. degree in electrical engineeringfrom the Federal University of Santa Catarina,Florianópolis, Brazil, in 2008.

Currently, he is a Professor with the FederalUniversity of Uberlândia. His research inter-ests include power electronics, control systems,renewable energy, and dc microgrids.

Fernando Lessa Tofoli received the B.Sc.,M.Sc., and Ph.D. degrees in electrical engineer-ing from the Federal University of Uberlândia,Uberlândia, Brazil, in 1999, 2002, and 2005,respectively.

Currently, he is a Professor with the FederalUniversity of São João del-Rei, São João del-Rei, Brazil. His research interests include high-power factor rectifiers, high-voltage gain dc–dcconverters, and renewable energy applicationsinvolving power electronics.

Ivo Barbi (M’78–SM’90–F’11) was born inGaspar, Brazil, in 1949. He received the B.Sc.and M.Sc. degrees in electrical engineeringfrom the Federal University of Santa Catarina,Florianópolis, Brazil, in 1973 and 1976, respec-tively, and the Dr.Ing. degree in electrical engi-neering from the Institut National Polytechniquede Toulouse, Toulouse, France, in 1979.

He is currently a Visiting Professor withthe Department of Automation and Systems,Federal University of Santa Catarina. He

founded the Brazilian Power Electronics Society and the PowerElectronics Institute of the Federal University of Santa Catarina.

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