Electrical Power and Energy Systems 33 (2011) 715–720

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Electrical Power and Energy Systems

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

Single-phase power supply to balanced three-phase loads throughSVAr compensators

F.R. Quintela ⇑, R.C. Redondo, J.M.G. Arévalo, N.R. MelchorEscuela Técnica Superior de Ingeniería Industrial, Universidad de Salamanca, 37700 Béjar, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 October 2008Received in revised form 7 December 2010Accepted 9 December 2010Available online 26 January 2011

Keywords:Static VAr compensatorSingle-phase power supplyThree-phase load

0142-0615/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.12.028

⇑ Corresponding author. Tel.: +34 923408080.E-mail addresses: felixrq@usal.es (F.R. Quintela), rob

jumagar@usal.es (J.M.G. Arévalo), norber@usal.es (N.R

Static VAr Compensators (SVCs) can transform phase-to-phase loads into balanced three-phase loads.This paper shows that this function of SVCs is reversible; that is, SVCs can be used to supply power froma single-phase line to balanced three-phase loads. Those three-phase loads then show themselves to thesingle-phase line as single-phase loads with the desired power factor, even unity.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Static VAr compensators (SVCs) are traditionally used to com-pensate reactive power in order to reduce energy losses in the lineand adjust voltages. Research aimed to optimize this function ispractically constant [1–5]. But, at the same time, SVCs can trans-form unbalanced three-phase loads into balanced three-phaseloads with any desired power factor, even unity. The result doesnot alter the active power extracted from the power system[6–9]. This transformation can also be applied to phase-to-neutralor phase-to-phase loads. SVCs transform both types of loads intobalanced three-phase loads without altering the active power ex-tracted from the power grid.

Until now, approaches to SVC analysis have been performedexclusively considering proper loads, that is, loads that absorb po-sitive active power. Some authors have even considered onlyimpedance loads, therefore limiting SVC application to just passiveloads [7–9]. Power analysis procedures plainly show [6], however,that SVCs can be as efficient when coupled to active loads as theyare when connected to passive loads, even if the power absorbedby those active loads is negative. This conclusion suggests thatthe functions of SVCs can be reversible. For example, a single-phasepower supply can be transformed into a balanced three-phase sup-ply that delivers power to a balanced three-phase load. In this case,it could be possible to supply power to balanced three-phase loadsfrom a single-phase line through the proper SVC or, in other words,

ll rights reserved.

errm@usal.es (R.C. Redondo),. Melchor).

a SVC could transform a single-phase line into a balanced three-phase power supply in order to deliver power to a balancedthree-phase load, which can be useful wherever the only energysource is a single-phase line (e.g., many rural areas) [10].

This paper shows that, in fact, this kind of power supply is possi-ble, and deduces the calculation procedure for the adequate SVC. Butfirst, some SVC related concepts that are used below are outlined.

2. Compensators

Fig. 1 shows a three-phase three-wire load with three identicalmeters. The readings of the meters are the active and reactive pow-ers the load absorbs through the phases [11,12]. As Kirchhoff’s firstlaw imposes the restriction that all the phase currents add up tozero, the following restrictions on the powers appear when thevoltages are balanced [6]:

2PR1 � PS1 �ffiffiffi3p

Q S1 � PT1 þffiffiffi3p

Q T1 ¼ 0

2Q R1 � QS1 þffiffiffi3p

PS1 � Q T1 �ffiffiffi3p

PT1 ¼ 0ð1Þ

If QR1 = QS1 = QT1 in (1), then it results that PR1 = PS1 = PT1 and theload is balanced. So, it suffices to modify the reactive powers theload absorbs through the phases to alter its balanced or unbal-anced state. Fig. 2 shows a procedure to achieve this, consistingin connecting a load, which will be called compensator, in parallelwith the load. If the reactive powers absorbed by the compensatorthrough its phases are Q C

R ¼ �Q R1;QCS ¼ �QS1, and Q C

T ¼ �Q T1, thatis, exactly the opposite of those absorbed by the load, then thereactive powers absorbed by the compensator-load set will beQR ¼ QC

R þ QR1 ¼ 0;Q S ¼ QCS þ QS1 ¼ 0, and Q T ¼ Q C

T þ Q T1 ¼ 0. The

P ,Q

Load••

R1

S1

T1•

•

•• R1

P ,QS1

P ,QT1

IR_

IS_

IT_

R

S

T

Fig. 1. Powers absorbed by a three-phase three-wire load through each phase.

Load

R

S

T

,QR1

,QT1

,QS1

QTCQS

CQRC

Compensator

PR1

PT1

PS1

,QR

,QT

,QS

PR

PT

PS

Fig. 2. If QCR ¼ �QR1, QC

S ¼ �QS1, and QCT ¼ �QT1, then QR ¼ QS ¼ QT ¼ 0, and the

compensator-load set is balanced with unity power factor.

716 F.R. Quintela et al. / Electrical Power and Energy Systems 33 (2011) 715–720

compensator-load set would then be balanced with unity powerfactor, because the reactive power extracted by the whole set fromthe power system is Q = QR + QS + QT = 0 [11].

Moreover, if the compensator does not absorb active power,then the load would have been balanced without altering the ac-tive power obtained from the system. This is possible by usingreactance compensators, also known as Static VAr Compensators(SVC).

3. Reactance compensators

Fig. 3 shows three delta-connected reactances, whose reactivepowers are QRS, QST, and QTR. The three-terminal network they formwill be called reactance compensator. The reactive powers it ab-sorbs through each phase are [6]:

Q CR ¼

QRS þ Q TR

2

Q CS ¼

QST þ Q RS

2

Q CT ¼

QTR þ Q ST

2

ð2Þ

Regardless of the values adopted for QCR , Q C

S , and QCT , there al-

ways are three delta-connected reactances that absorb those reac-tive powers through the phases. These three reactances are thesolution of (2):

Q RS ¼ QCR þ Q C

S � Q CT

Q ST ¼ Q CS þ Q C

T � Q CR

Q TR ¼ QCT þ Q C

R � Q CS

ð3Þ

RSQ QST

TRQ

RQC QS TQC C

R S T

Fig. 3. Reactance compensator. The right values of QRS, QST, and QTR define acompensator that absorbs any desired reactive powers through the phases.

Therefore, it is always possible to design compensators that ab-sorb the desired reactive power through each phase. Seeing thatthese compensators do not absorb active power, because they areformed only by reactances, they can be used to balance anythree-phase three-wire load without modifying the active powerextracted from the power system. For example, in order to balancethe load depicted in Fig. 2 while achieving unity power factor, itsuffices to connect a compensator that absorbs through each ofits phases exactly the opposite reactive powers the load absorbsthrough theirs, that is, QC

R ¼ �QR1;QCS ¼ �QS1, and QC

T ¼ �QT1.The reactive powers of the needed reactance compensator are ob-tained by substituting those reactive powers in (3):

Q RS ¼ �QR1 � Q S1 þ Q T1

Q ST ¼ �Q S1 � Q T1 þ Q R1

Q TR ¼ �Q T1 � Q R1 þ Q S1

ð4Þ

Eq. (4) give the powers of the delta-connected reactances; thesepowers depend on the reactive powers absorbed through thephases by the three-phase load. The result is that the reactivepower absorbed by the whole compensator-load set is zero, anda balanced load that absorbs no reactive power is obtained. Shoulda balanced load that absorbs a reactive power Q be preferred, thenQ/3 must be added to the expressions on the right side of the equalsign of (3) and (4) [6].

4. Transforming phase-to-phase loads into balanced three-phase loads

Reactance compensators can be employed to transform phase-to-phase loads into balanced three-phase loads. The readings ofthe identical var-meters depicted in Fig. 4a give the reactive powerthat the load connected to phases R and S absorbs through thosephases [11,12]. The reactive power absorbed by the load throughphase T is zero: QT1 = 0. If those values are substituted in (4), thenthe reactances of the compensator that transforms the load con-nected to phases R and S into a balanced three-phase load withunity power factor are obtained:

Q RS ¼ �QR1 � Q S1

Q ST ¼ Q R1 � Q S1

Q TR ¼ �Q R1 þ Q S1

ð5Þ

The reactances of the compensator can also be obtained fromthe readings of the meter in Fig. 4b, which gives the active andreactive powers of the load. Seeing that PT1 = QT1 = 0, thenP1 = PR1 + PS1, and Q1 = QR1 + QS1 [11]; from (1):

Q R1 ¼ �1

2ffiffiffi3p P1 þ

12

Q1

Q S1 ¼1

2ffiffiffi3p P1 þ

12

Q 1

ð6Þ

Substituting these values in (5):

(b)(a)R

S

T

PR1•

•

PS1•

•

O

,QR1

,QS1

PT1•

• ,QT1

R

ST

P1•

•

QTR

QRS QST

,Q1

Fig. 4. (a) Three identical meters placed to measure the active and reactive powersof each phase. (b) A reactance compensator can transform a phase-to-phase loadinto a balanced three-phase load with the desired power factor, even unity.

F.R. Quintela et al. / Electrical Power and Energy Systems 33 (2011) 715–720 717

QRS ¼ �Q 1

QST ¼ �P1ffiffiffi

3p

QTR ¼P1ffiffiffi

3p

ð7Þ

The expressions show that if the active power P1 and the reac-tive power Q1 of the phase-to-phase load are known, then the com-pensator reactances that transform the former load into a balancedload with unity power factor can be obtained. If the compensator-load set is required to absorb any desired reactive power Q whileremaining balanced, then (7) has to take the form:

QRS ¼ �Q 1 þQ3

QST ¼ �P1ffiffiffi

3p þ Q

3

QTR ¼P1ffiffiffi

3p þ Q

3

ð8Þ

wherein Q/3 has been added to the expression on the right side ofeach equal sign of (7).

5. Single-phase power supply to balanced three-phase loads

Whenever the voltages of the three-phase power system arebalanced, the variables involved in (8) must fulfil the relationsshown in those formulas. Therefore, any three of those variablescan be arbitrarily fixed, and the other three are deduced from(8). If a negative value is given to P1, then the load is a generator.It is possible to set the value of Q1 too: if Q1 = 0, for instance, thenthat generator neither absorbs nor delivers reactive power. Asthere still remain four variables, one can still be arbitrarily fixed.

Fig. 5a is similar to Fig. 4b, but now the load connected tophases R and S is explicitly shown as a generator. The effectivevoltage of that generator is U = URS, where URS is the phasor of thatvoltage. If Pg is the active power delivered by that generator, thenP1 = �Pg. Furthermore, Pg is equal to the active power PL absorbedby the balanced three-phase load placed to the left side ofFig. 5a, as the compensator neither absorbs nor delivers activepower because it is formed only by reactances. It is also true thatQ1 = �Qg, where Qg is the reactive power delivered by the genera-tor. In (8) Q = �QL, where QL is the reactive power absorbed bythe three-phase load on the left side of Fig. 5a. Generally, Q g–Q L,as the compensator can deliver or absorb reactive power. If thosevalues are substituted in (8), then it results:

QRS ¼ Qg �QL

3

QST ¼PLffiffiffi

3p � Q L

3

QTR ¼ �PLffiffiffi

3p � Q L

3

ð9Þ

R

ST

P1•

•

QTR

QRS QST

(a),Q1

URS

_+Three-phaseloadP ,Q

(b)

L L

I_I_

Fig. 5. (a) Transformation of a single-phase generator into a three-phase generator. (b)figure. The network represents the power supply from a single-phase line to a balanced

Eq. (9) give the powers of the compensator reactances that al-low a certain single-phase generator to deliver the power Pg = PL

to a balanced three-phase load. The single-phase generator deliversto the compensator-load set the reactive power Qg, which is gener-ally just a part of the reactive power QL absorbed by the load. In-deed, by adding the three formulas of (9) and solving Qg, itresults that Qg = QL � (�QRS � QST � QTR). As �QRS � QST � QTR isthe reactive power delivered by the compensator, it follows thatthe generator only delivers the reactive power not supplied bythe compensator to the load. This distribution of reactive powersonly influences the value of QRS, which is the sole compensatorreactance that depends on Qg.

As stated above, it is possible to fix any desired value for Qg in(9). For example, Qg = QL, so the addition of (9) givesQRS + QST + QTR = 0; that is to say, the compensator does not absorbnor deliver reactive power, and all the reactive power absorbed bythe load is delivered by the single-phase generator. An interestingfact is that whenever the compensator absorbs neither active norreactive power, its phase currents form a balanced negative-phase-sequence set [6].

A better solution is to design the compensator so that the gen-erator does not have to deliver nor absorb reactive power. It suf-fices for that purpose to make Qg = 0 in (9). The compensator willthen deliver all the reactive power QL absorbed by the load. In ef-fect, if the three formulas of (9) are added with a sign change,and with Qg = 0, then the result is �QRS � QST � QTR = QL. The firstmember of this expression is the reactive power delivered by thecompensator, which is equal to the power absorbed by the load.

Fig. 5b is a more convenient representation of Fig. 5a. The sin-gle-phase generator is now supposed to be on the left side of thefigure, though only the single-phase line coming from it has beendepicted. The compensator allows to supply the active power PL

and the reactive power QL to a balanced three-phase load from thatsingle-phase line. In order to design such a compensator it sufficesto know the powers PL and QL absorbed by the three-phase loadwhen connected to a three-phase line of balanced voltages, where-in the effective phase-to-phase voltage is U. Eq. (9) provide thecompensator reactances, and U coincides with the effective voltageof the single-phase line.

The amount of the reactive power Qg that will be delivered bythe single-phase line can be arbitrarily fixed. As has already beensaid, an appropriate value could be Qg = 0, because then the sin-gle-phase line does not supply any reactive power, therefore pro-ducing the smallest possible power loss in the line. In otherwords, from the point of view of the single-phase line, the compen-sator designed using (9), and Qg = 0 as restriction, transforms thethree-phase load into a single-phase load with unity power factor.Thus, the effect achieved by the reactance compensator may alsobe interpreted as a transformation of a balanced three-phase loadinto a single-phase load with, at the same time, its power factorcorrected to any desired value, even unity. Note that this transfor-

R1Three-phaseloadP ,Q

Pg•

• ,Qg

URS

QTR

QRS QST

L L

I_

S1I_

T1I_

SI_

C

TI_

CRIC

_

_

Generalization of (a) where the three-phase load is placed to the right side of thethree-phase load.

R1

URS

I_

S1I_

T1I_

SI_C

TI_C

RIC

_

_

Z_Z_Z_

O

QTR

QRS QST

IRS

_I

TR

_

I

ST_

T

S

R

Fig. 6. Power supply from a single-phase line to a balanced three-phase load.

718 F.R. Quintela et al. / Electrical Power and Energy Systems 33 (2011) 715–720

mation does not affect the performance of the load, which contin-ues to work as a balanced three-phase load.

The values of the compensator reactances can be obtained fromtheir reactive powers by taking into account that, in general:

X ¼ U2

Qð10Þ

wherein X is any reactance that absorbs the reactive power Q whenits effective voltage is U.

6. Direct analysis

All of the above relies on the analysis of SVCs, and leads to thenetwork of Fig. 5b. In this network, a compensator allows to supplyelectric power to a balanced three-phase load by means of a single-phase line. This reasoning seems quite convenient because it sug-gests the reversibility of SVCs. However, a direct analysis ofFig. 5b can be implemented without any reference to SVCs. The val-ues of QRS, QST, and QTR can be calculated so that the phase voltagesand currents in Fig. 5b be balanced. The effective voltage U = URS,the active power PL, and the reactive power QL absorbed by the bal-anced three-phase load, would be the starting values. The reactivepower Qg, which the single-phase line has to deliver, must be fixed,too. Qg can amount to any value, including zero. Once all these dataare fixed, the complex power delivered by the single-phase line isSg ¼ PL þ jQ g . The complex power absorbed by the balanced three-phase load is SL ¼ PL þ jQL, and the power absorbed through eachphase is S ¼ ðPL=3Þ þ jðQ L=3Þ. Therefore, the phase currents are:

IR1 ¼S

VR

!�

IS1 ¼S

VS

!�

IT1 ¼S

VT

!�ð11Þ

wherein VR ¼ V=0�, VS ¼ V=� 120�, and VT ¼ V=120�, withV ¼ U=

ffiffiffi3p

. The asterisk � denotes the conjugate of the complexnumber. Only two of the three equations in (11) are independent,as the currents add up to zero.

The currents of the three compensator terminals are:

ICR ¼ �j

QRS

U�RS

þ jQ TR

U�TR

ICS ¼ �j

QST

U�ST

þ jQ RS

U�RS

ICT ¼ �j

QTR

U�TR

þ jQ ST

U�ST

ð12Þ

wherein URS ¼ VR � VS, UST ¼ VS � VT , and UTR ¼ VT � VR. Onceagain, the three equations in (12) are not independent of each other,as the three currents add up to zero too.

The current of the single-phase line is:

I ¼ Sg

URS

!�ð13Þ

Moreover:

I ¼ IR1 þ ICR

� I ¼ IS1 þ ICS

IT1 ¼ �ICT

ð14Þ

If the currents in (14) are replaced by their respective valuesfrom (11)–(13), then three complex equations or six real equations

are obtained. Only three of those real equations are independent,and can lead to obtain the expressions deduced in (9) just by solv-ing the values of QRS, QST, and QTR, that is, just by obtaining the reac-tance powers of the compensator that allows to deliver electricpower to the balanced three-phase load through a single-phaseline.

7. Example

Let a three-phase load be formed by three identical impedanceswhose values, in ohms, are Z ¼ Rþ jX ¼ 4þ j3, and let there be asingle-phase line of effective voltage U ¼ 400 V. A balancedthree-phase power source is created for that load (Fig. 6) by meansof a SVC, so that the phase voltages are

VRO ¼ VR ¼Uffiffiffi3p =0�

VSO ¼ VS ¼Uffiffiffi3p =� 120�

VTO ¼ VT ¼Uffiffiffi3p =120�

The effective current through each phase will then be IR1 ¼ IS1 ¼IT1 ¼ I1 ¼ V=Z ¼ 80=

ffiffiffi3p

A, wherein V ¼ U=ffiffiffi3p

, and Z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi42 þ 32

p.

The active power absorbed by the three-phase load is PL ¼ 3RI21 ¼

25600 W, and the reactive power is Q L ¼ 3XI21 ¼ 19200 VAr. If

these data and Qg = 0 are used in (9), then the reactance powersof the compensator, in VAr, are QRS = �6400, QST ¼ �6400þ25600=

ffiffiffi3p

, and Q TR ¼ �6400� 25600=ffiffiffi3p

. The compensator deliv-ers all the reactive power that the load needs, as �QRS � QST � QTR =19200 = QL. The single-phase line does not deliver reactive power.The values of the compensator reactances can be calculated using(10).

If it is preferred that all the reactive power be delivered by theline instead, it suffices to make Qg = QL. From (9) it results thatQRS = 12800, and the other two reactance powers remain un-changed. Now, the compensator is neither providing nor absorbingany reactive power. Indeed, QRS + QST + QTR = 0. All the reactivepower absorbed by the three-phase load is delivered by the sin-gle-phase line. The compensator phase currents are, in this case,as follows:

ICR ¼ IRS � ITR ¼

�jQ RS

U�RS

��jQ TR

U�TR

¼ 46:19=23:13�

ICS ¼ IST � IRS ¼

�jQ ST

U�ST

��jQRS

U�RS

¼ 46:19=143:13�

ICT ¼ ITR � IST ¼

�jQ TR

U�TR

��jQST

U�ST

¼ 46:19=� 96:87�

IR

IS

IT

_

_

_ Load

VR

_VS

_VT

_

URS

_R

S

T

(a)

Load

IR

IS

IT

_

_

_

VR

_VS

_VT

_

URS

_ R

S

T

(b)

SVC

Fig. 7. Feeding a three-phase load from (a) a three-phase line and (b) a single-phase line through the appropriate SVC.

F.R. Quintela et al. / Electrical Power and Energy Systems 33 (2011) 715–720 719

which form a balanced negative-phase-sequence system, as men-tioned above, because the compensator neither absorbs nor deliversreactive power.

8. Experimental tests

Computer simulations that corroborate the above results weredeveloped using Mathematica (Wolfram Research, Inc.) and Lab-VIEW (National Instruments, Corp.).

Also, a three-phase load, which was formed by three identicalY-connected impedances, was first coupled to a three-phase powerline (Fig. 7a) and its phase voltage and currents were measured

Fig. 8. Screenshot of the phase voltages (top) and the phase currents (bottom) ofthe three-phase load that is connected to the three-phase power grid. Labels overeach waveform are their effective value.

Fig. 9. Screenshot of the phase voltages (top) and the phase currents (bottom) ofthe three-phase load connected to a single-phase line through the appropriate SVC.

using a National Instruments DAQ-Card and a Macintosh computerrunning LabVIEW [6]. Then that load was connected to a single-phase line using the appropriate SVC (Fig. 7b). The single-phaseline of Fig. 7b was formed by two of the three phases of the formerthree-phase line. The phase voltages and currents were once moreobtained. The results of both configurations were as expected: thephase voltages and currents measured when using a single-phasepower supply to feed the three-phase load through the SVC werethe same of those obtained using the three-phase power supply.

Fig. 8 shows the voltage and current of each phase when theload was connected to the three-phase power grid, and Fig. 9shows those voltages and currents when the load was connectedto the single-phase line through the SVC. These results clearlyshow that the load could be fed as a balanced three-phase loadfrom the single-phase line, with all its phase voltages and currentsbalanced.

Finally, in order to avoid the stroboscopic effect, a balancedthree-phase voltage system obtained with a SVC from a 230 Vphase-to-neutral line was used to feed three discharge lamps in adelta connection (Fig. 10). The SVC was also designed so that thewhole set had unity power factor. Fig. 11 shows the balanced

uR

N

i

uRS

uSTuTR

1.5 F3.8 F

7.5 F

µµ

µ

Fig. 10. Delta-connection of discharge lamps.

Fig. 11. (a) Balanced three-phase voltage system that prevents the stroboscopiceffect. (b) Voltage and current in the single-phase line.

720 F.R. Quintela et al. / Electrical Power and Energy Systems 33 (2011) 715–720

three-phase voltages of the lamps, and the voltage and current ofthe single-phase line. The power factor of the whole set is 0.99.Due to the initially low power factor of the lamps (0.4), the SVCcould be formed only by three capacitors of 7.5, 3.8, and 1:5 lF.

9. Conclusions

This paper shows that SVCs can be used to obtain a balancedthree-phase system from a single-phase line in order to supplypower to a balanced three-phase load. This feature can be usedto incorporate SVCs in the design of three-phase receivers thatcould be connected wherever single-phase lines are the only onesavailable. A SVC whose values are automatically adapted to everyload can also be the basis of a phase converter used to feed variablethree-phase loads from single-phase power supplies, so that theoverall power factor remains near unity. Formulas for the designof the SVCs based on various hypotheses have been deduced.

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