Harmonic Power Flow for Unbalanced Systems

7/27/2019 Harmonic Power Flow for Unbalanced Systems

http://slidepdf.com/reader/full/harmonic-power-flow-for-unbalanced-systems 1/8

2052 IEEE Transactionson Power Delivery, Vol. 8,No.4,October 19'93

HARMONIC POWERFLOW FOR UNBALANCED SYSTEMS

Manuel ValciuCel Julio G. Mayordomo

Universidad PolitkNca de MadridE.T.S. de Ingeniems Industriales

Departamentode Ingenieria El&tricaJo& Gutibrrez Abascal,2 28006 Madrid, Spain

ABSTRACT.- In this paper a harmonic power flow thatanalyzes harmonics in unbalanced systems is presented .Thedeveloped algorithm has two steps which are executed

successively: the first is a fundamental frequency power flowfor the ac linear network in which non-linear loads arerepresented by current sources. The second is a frequency-domain iterative Newton-Raphson method to calculate theharmonics generated by non-linear loads. In this second step,the ac linear network is represented by a generalized Thevenin

equivalent with respect to the non-linear loads, obtained fromthe power flow solution. Both linear and non-linear loads are

considered in terms of power.I"lectrical networks are normally unbalanced, and they have a

certain degree of imbalance that depends on the network

composition and operation. Balanced operation is usuallyassumed and then many simplifications in its representationand study can be considered. However, in certain situations it isimportant to take into account the imbalances and their

influence on thegeneration of non-characteristic harmonics dueto three-phase non-linear devices [13].

Xia and Heydt [l], and other authors (references [21 to [51),developed a harmonic power flow for balanced systems, wherelinear and non-linear loads are treated in terms of power. Thismethod is a reformulation of th e conventional Newton-Raphson power flow method to include non-linear loads. It is

based on the simultaneous resolution of the harmonic balance

and power constraint equations in all buses.

Several harmonic load flows have been recently described for

balanced systems [16] and for unbalanced systems (references

[13] to [15]), that solve the harmonic interaction by means ofa Gauss algorithm. The main drawback of this algorithm incomparison with the Newton methods [ 6 ] , 7] is its limited

convergence capability, and difficulty in adjusting control

variables of non-linear loads (e.g. firing angle and d.c. currentin a converter).

93 WM 061-2 PWRD A paper recommended and approvedby the IEEE Transmission and Distribution Committee

of the IEEE Power Engineering Society for presenta-

tion at the IEEE/PES 1993 Winter Meeting, Columbus,OH, January 31 - February 5, 1993. Manuscript sub-

mitted March 2, 1992; made available for printingNovember 3, 1992.

The purpose of this paper is to present a harmonic power flowprogram which allows the calculation of the harmonic voltagesand currents in a three phase balanced or unbalanced network

with distributed converters. The developed algorithm [91 hastwo steps which are executed successively: the first step is afundamental power flow for the ac linear network in whichnon-linear loads are represented by current sources. In thesecond step a frequency-domain iterative Newton-Raphsonmethod calculates the harmonics generated by n on-l iw loads.In this second step, the ac linear network is represented by ageneralized Thevenin equivalent with respect to the non-linear

loads, obtained from the power flow solution. In both steps,loads are considered in terms of power.As it is shown in thispaper, the use of the Newton-Raphson method allows thesolution of strong harmonic interactions at non-Characteristic

harmonics

The authors have developed a harmonic power flow for balancedsystems [8] which is extended here to unbalanced systems.

HARMONIC POWER FLOW P R OC E D U M

The harmonic power flow program RCADE consists of twosubprograms executed successively and iteratively:ARMO-D(Harmonic Analysis) and RCFD (Unbalanced power flow).

Harmonic wwe flow w o n method

The flow chart of the harmonic power flow is depicted in figure

1,

Power Flow RCFD

Itel=Iter+l

Figure 1 Flow chart of RCADE.

0885-8977/931$03.00 993 EEE



2053

The procedure starts reading the data file (1). Step (2)consists

of the unbalanced power flow RCFD at fundamental frequency.

Once the solution is reached, conventional loadsare replaced byimpedances in order to obtain the generalized Thevenin

equivalent. This Thevenin equivalent is reduced to buses withnon-linear loads, and it is used in step (3). A harmonic analysis

is then performed (ARMO-D)olving the harmonic balanceequations and adjustingthecontrol variables of non-linear loadsaccordingto their power constraints.

In step (4) the fundamental frequency sequence voltagesobtained in (2) and (3) at the non-linear buses are compared. Ifthey are not equal (or in practical terms, if their differences are

not less than a tolerance), the fundamentalfrequency sequencecurrents obtained in (3) become the new current specifications

in (2) for non-linear loads. This process is repeated until the

solution is reached and the results are printed.

MO-D)

The program ARMO-D [6,7] is based on a frequency-domain

iterative method. It computes accurately the steady-statevoltages and currents generated by a converters operation in athree-phase unbalanced network.

This program allows the simulation of distributed converters

by means of harmonic balance technique with good

convergence. In this program the ac linear network is reduced toa generalized Thevenin equivalent with respect to the non-linear

loads. This Thevenin equivalent is obtained by gaussianreduction of the Ybus matrix formed at each harmonicfrequency from the linear elements models; conventional loads,

lines, capacitors, filters, etc.

Conventional oads can be modelled by a seriesR-L mpedance,parallelR-Lmpedance,or by a specified combination of both.

Non-linear loads are modelled by voltage-controlled current

sources. In the simplest case (an 1-phase power system with a

non-linear load) this behaviour is represented by:

i(t) = i (U) (1)

In the frequency domain, equation (1) becomes,

Ik = Ik (Ul, ...,Uk, ..., u m , ...)

which together with the linear network Thevenin equivalent,

k k k kE = U - Z I (3)

provides in theory a set of infinite non-linear equations, where:

k,m :harmonic order.k k k

Ik :Phasor of harmonic current (I = 1, +j Ix ).

k kUk :Phasor of harmonic voltage (Uk=U /e ).

Ek :Phasor of harmonic voltage (open circuit).

Zk :Driving point impedance.

F~ :Harmonic voltage error function (F = F~ +j F~ 1.k k

The harmonic analysis is performed with a truncated Fourier

series. Assuming a finite harmonic number h, expressions (2)and (3) yield a set of 2h non-linear equations (real andimaginary parts of harmonic voltage e m r €unction).For theiteration "w",theseequationare represented by:

Fkw=U w-(E - k kI w )(4)

The harmonic voltage balance between linear and non-linearnetwork is reached when

Fk = 0, k = 1,...b (5 )

The Newton-Raphson algorithm to solve (5 ) has the following

expression:

where :

[ F ]

[ U ]

[ J ] Jacobian matrix.

Error function vectorQ = 1, ...,h).Harmonic voltage vector (k= 1, ...,h)

The terms of the Jacobian matrix represent the coupling

between harmonics in each iteration. Thesearecalculated from

the sensitivities of the harmonic currents Ik with respect to the

harmonic voltages Um.

This formulation can easily be extended to balanced three-phasesystems with np distributed converters in several buses by a

generalized Thevenin equivalent, reduced to the buses with non-

linear loads. In these conditions 2nph non linear equations

must be solved.

For unbalanced three-phase systems, the number of equations is

6nph. However, many three-phase converter configurations

present no paths to the zero sequence currents. Under these

conditions, only the positive and negative harmonic currents

need to be solved, namely, 4nph equations. Thus, equation (4)becomes:

where the subscript"s" enotes sequence magnitude (1 or2).

For these reasons, the computer program has been developed inharmonic sequence quantities.

The basic converter configuration is a six pulse converter. Thisis formed by a Graetz bridge and an isolation transformer.Thedc side converter current Id is assumed perfectly smooth. The

secondary transformer ac side is represented by its commutating

voltages and short circuit impedances. Once defined the controlvariables Id and (firing angle), the waveform of the phase

currents are calculated from: commutating voltages; zero

crossings of the commutating voltages; overlap angles;transformer impedances and control variables Id and%. Zerocrossings and overlap angles are determined numerically usingan iterative procedure (Newton-Raphson).



2054

The unknowns in ARMO-D program are positive and negative

harmonic voltage sequence components, U 1k. U2k in the

primary side of the converter transformer and the controlvariables ( Id, ao).From the values of these unknowns in a

ARMO-D w-iteration, the corresponding harmonic current

sequence components are calculated. It can be expressed in

analytical form by,

where:

subscripts 1 and 2 represents positive and negative

~ U € Z l W .

k kU1 and U2 are the sequence magnitudes

corresponding o phase ones.

The relationship (8) is an extension of the Xia and Heydtconverter model [ l] to unbalanced systems. A detaileddescription of equation (8) is given in references [6]and [9].The way in wich converter harmonic currents, and sensitivities

are obtained is briefly described. The following steps must betaken:

Step 1.

Step 2.

Step 3.

Step 4.

step 5

Determination of the harmonic voltage

sequence components in the secondary of the

transformer from the primary harmonic voltages.Transformer phase shiftsmust be considered.

Transformation of the harmonic voltages in phasequantities, and determination of the 6 voltage zero

crossings and 6 overlap angles.

Determination of phase harmonic currents from

the waveform of phase currents.

Transformation of phase harmonic currents in

sequence magnitudes, and determination of thesensitivities of harmonic currents with respect to the

harmonic voltages and to the control variables.

Adaptation of harmonic currents and their sensitivities

to the primary ac side of the transformer. When thereare several converters connectedto the Same bus, the

total harmonic currents and their sensitivities are

obtained by summation of the individualcontributions.

In the absence of firing angle errors and of background evenharmonics voltages, half wave symmetry exists in phasevoltages and currents. In these conditions onlv 3 zero

crossings, 3overlap angles and the odd harmonic currents need

to be solved.

Once the harmonic currents and their sensitivities have beencalculated for each non-linear load, the4nph harmonicbalance

equations and their sensitivitiesare obtained by substitution ofequation (8) into (7). However, it is necessary to include 2np

new error functions ( 2 per non-linear load) to define theoperating point of converters in terms of active power P and

apparent power S. By adjusting the control variables it ispossible to satisfy thesepower constrainterror functions,givenfor a converter by:

v - - - - k=1

Fp =' 'scheduled

A A

FS=StL 't scheduled

(9)

Total active power P is defined, consideringh harmonics, as:

h

P=Real[3 C ( U1 k I1k * +U I t * ) ] (11)

k = IAlthough there is not any internationally accepted definition for

apparent power in unbalanced systems, the definition adopted

by Depenbrock [121 is considered here.

Apparent power St, for an ungrounded load, isdefined as:

where:I,, I and IC are the rms values of phase currents.

Ua ,u b and Uc are the rms values of phase voltages

without zero sequence.

the final expression, in sequence components,is:

S t 9 (U12 +U ( 112+ b 2 ) (13)

Considering h harmonics, the currents and voltages of thisexpression become:

h hU+ ( Ulk)2 ; U+ ( u2 2

k=1 k= 1

h

112= C [ ( Irlk l2+ ( Ixlk )* 1

k=

h



2055

The sensitivities of error functions Fp and Fs to harmonic

voltages and control variables are obtained from equations (1 1)and (13) taking into account the sensitivities of harmonic

currents of equation (8).

Program ARMO-D uses the Newton-Raphson algorithm to

solve the 4nph harmonic balance equations and 2np power

constraint error functions adjusting simultaneously all

unknowns: 4nph harmonic voltage sequence components and2np control variables.

LlAdawd power flow subproeram lRcFDL

As mentioned before, it is a Newton-Raphson power flow

program for fundamental frequency, compatible with ARMO-Dsubprogram that solves the three sequences considering threetypes of buses: three-phase sources, conventional loads andnon-linear loads.

The considered conventional loads are ungrounded and have adelta connection. Two options are included: structurallybalanced load and structurally unbalanced load. In the first case,only total active and reactive powers are scheduled, whereas in

the second caseactive and reactive powers of each delta branch

need tobe scheduledas shown in the Appendix.

Non-linear loads are considered as fundamental frequencyconstant current SO^,obtained !?om ARMO-D program.

Three-phase sources are represented by means of a reactance

with the scheduled positive and negative sequence voltages (in

magnitude and angle) behind this reactance.

The sequence currents of a conventional load in a bus p havethe next form, in complex magnitudes,

(15)1L1(U , U 2, bus power scheduled , ype)

I L2(U , U 2’ bus power scheduled , type)

P

Conventional load currents are functions of the type of load,bus voltages and power specifications. The appendix containsthe current expressions for different types of conventional

loads. The non-linear load currents are calculated in Subprogram

ARMO-D, and they are considered in subprogram RCFD as

independent currentsources.

The error functions corresponding to fundamental frequencycurrent balance at bus p are, in complex magnitudes,

where: 1,2,0 :positive, negative and zero sequence,p, q :bus index.

The non-linear current balance equations and their sensitivities

are obtained by substitution of equations (15) into equations

(16). However, the conventional load currents of equations (15)

do not depend on the zero sequence voltage. Therefore, it is

possible to eliminate linearly the zero sequence voltage fromthe positive and negative current balance equations. Thereforeprogram RCFD solves only current balance equations ofsequences 1 and 2 using Newton-Raphson aleorithm. Zerosequence voltages are obtained linearly from the resultingpositive and negative sequence voltages.

EXAMPLE

The test system presented here is a 8 bus network taken fromreference [ll], with a 6-pulse converter located at bus 8, asshown in figure 2.

All loads are structurally balanced except thecorrespondingbus4, whose value is 4.2 MVA, cos (p=0.8. connected between

phases A and B. Loads are modelled as R-L parallelconfiguration. Table 1 shows the network data in p.u.

The source has only scheduled positive sequence voltage, equalto 1.0 pu. In parallel to a 4 MVAr capacitor at bus 8, filters

tuned at 5th, 7th, 1 * and 13* harmonicsare connected (eachof the first two filters has a value of 1.8 MVAr and each of theremaining two filters has a value of 0.225 MVAr, all of themwith quality factor of 100). The study includes the first fifteenodd harmonics (h=15).

SCC 410MVA

Figure 2. Example.



2056

Bus A-----------

ONEONEONETHREE

THREE

FOURFIVE

ONE

B u s B R X G B

TWO .00149 . OW63 .00000 . WO38FOUR .01220 .02916 .00OOO .00080SEVEN .00489 .01169 .00OOO . WO32FIVE .00243 .00259.00000 .01413SEVEN .00789 .01886.00000 .00051FIVE .00243 .00259 .00OOO .01413SIX .01239 .04056 .00OOO .00118SIX .00170 .00140.00OOO .02659

--_---_--- _-_--___--______-----_- _-----__

Table 1. Network data

The full solution was reached in two iterations of the complete

process according to the flowchart of figure 1. Tables 2and 3show the results of RCFD subprogram and the first ten odd

harmonic voltages at the most distorted bus (bus8)provided bythe ARMO-D subprogram. It is important to notice that theunbalanced load at bus4causes the generation of considerableamount of non-characteristic harmonic voltages. The highest

values are 13 % and 4.3% for 3rd and 9* harmonics

respectively.

Bu s

ONE

SEVEN

SIXFIVETHREE

TWO

FOUREIGHT

-_-_----

Ua(Mag,Ang) Ub(Mag,Ang) Uc(Mag,Ang)

_-__------_____-_________--______------_--------__-.9928 -3.54 .9792 -122.84 .9965 117.48.9905 -4.35 .9752 -123.50 .9956 116.84.9919 -3.57 .9783 -122.84 .9960 117.47.9810 -4.54 .9630 -123.40 . 9888 116.93.9819 -4.55 .9641 -123.45 .9891 116.89.9902 -3.72 .9766 -123.02 .9939 117.31.9809 -4.56 .9626 -123.33 .9899 116.971.0236 -6.7010076 -125.861.0286 114.49

Table2 RCFD Solution (2nd complete iteration).

Harm. Ua(Mag,Ang)-----___l--_______l-______1 1.02359 -6.70

3 BO246 -179.045 .00164 -117.317 .00083 -162.759 80724 169.3411 .00235 -81.2613 BO164 -126.5915 .OOO73 119.44

Ub(Mag,Ang) Uc(Mag,Ang)I------_----------------------------_-I1.00759 -125.86 1.02857 114.49

.01308 16.25 .01072 -160.29

.00170 6.32 .00158 126.38

. OW75 80.76 . WO83 -36.44

.04339 -35.67 .03696 139.58

.00242 47.13 .00208 164.86

.00132 116.77 .00158 5.22

.00332 -33.09 .00269 154.04

Firing angle: a= 1086155DC current : Id=0.98726pu.

Table3. ARMO-D Solution (2nd complete iteration). Bus 8.

The relations voltage/current for the 3rdharmonic are 1.062pu

for positive sequence and 1003pu for the negative sequence,

while for the 9ththey are 3372pu and 4.422pu respectively.These high values in impedance magnitudes indicate a strong

harmonic interaction condition at non-characteristic3rd and 9*

harmonics.

With respect to accuracy of the method, two items must bementioned the truncation effect and the determination of the

zero crossings of the line voltages.

The truncation effect was studied by executing the programRCADE with a different number of harmonics h. As shown n

figure 3, the consideration of the first 13 odd harmonics isenough to confirm the negligible truncation effect (the base

casewas executed with h=15).

Number of harma) Phase b

b) phase Number Ofh m .

Figure3. Truncation effect with program RCADE.

This example was analysed also with EMTP 1101obtaining agood agreement with RCADE for fundamental frequency andcharacteristic harmonics. Non-characteristic harmonicspresented discrepancies due to the fact that these harmonicsarevery sensitive to line voltage zero crossings. With EMTP

(time domain), small variations in the line voltage zerocrossings produce considerable changes in these harmonics.Therefore, a short time step is necessary (e.g. 10ps). Thisproblem does not appear in the frequency domain (programRCADE) because zero crossings are determined numericallywith very high precission.

Differences between RCADE and EMTP solutions are shown

in table4 or the non-characteristic harmonics3* and gth. The

highest difference is 6.295 for the 9* in U,.

RCADE 0.010720.01028

RCADE 0.04339 0.036960.04178 0.03479

Table4.Discrepancies between EMTP and RCADEprograms.

From base case (A), four modifications are studied:

B. Lines in layer disposition, in order to take into

account the structurally imbalance+of networklines.

C . Series R-L configuration of the unbalanced load.

D. System with two converters. The conventionalload at bus 2 s substituted by a converter with



2057

S = 4 MVA, P = 3.4 MW and Xtr = 5 %.

E. Series R-L configuration of the unbalanced load

in the situacion of case D.

As shown in table 5, the disposition of the line in lays doesnot change notably the harmonic distortion of the base case.However, the series R-L configurationfor the unbalanced load

decreases the damping of the harmonic impedance and thereforehigher distortion is produced: 2 % and 7.4 % for 3rd and gthharmonics respectively. Case D assumes the harmonicinteraction between two converters, rising the distortion of the

base case to 1.7 % and 5.2 % for 3rd and gth harmonics

resDectivelv. Case D is less severe than case C in suite of the

Table 6.b. 9* harmonic voltages for caseE.

CONCLUSIONS

existence bf two converters acting simultaneousiy on thenetwork. The joint effects of cases C and D provide the worst A new harmonicPOwer *Ow forunbalanced has been

situation, eading to a levelof % and 8 % for 3rd and 9thdescribed. It allows the analysis of characteristic and non-

characteristic harmonics, generated by converters (or others

non-linear elements) in their interaction with the utilityarmonics respectively.

network. he develop& procedure isbased on the ntegration fa fundamental frequency power flow subprogram and aniterative harmonic analysis subprogram, in which conventionaland non-linear loads are treated in terms of power. The modular

structure and versatility of both subprograms permit them to

take advantage of the developed software for the harmonicrepresentation of the linear network (harmonic penetration

programs), as well as to reduce the number of non-linearharmonic balance equations by using equivalents of the linear

network.

The fundamental frequency power flow is formulated in termsof Current balance equations for COnVentiOnd loads. This

formulation and the use of the Newton-Raphson algorithm, is

very suitable to deal efficiently with different load structuresand power specifications.

The iterative harmonic analysis subprogram also uses theNewton-Raphson algorithm to solve the harmonic balance andpower constraint equations of non-linear loads. This

subprogram allows the convergence under strong harmonic

interaction conditions, even when severe resonancesare present.

Moreover, this subprogram permits an easy adjustment of

converter control variables accordingto power constraints.

In all cases only two iterations of the complete process were

necessary, with a tolerance of OOOO1 pu .

Table 5.a. 3rd harmonic voltage at bus 8 for A to E.

Table 5.b. gth harmonic voltage at bus 8 for cases A to E.

Table 6 shows harmonic voltages for the non-characteristic 3rd REFERENCES

and 9* harmonics in all buses in the case of highest harmonic

distortion, caseE. [l ]D. Xia. G.T. Heydt,"Harmonic Power Flow Studies. Parts I& 11." IEEE Trans on PAS, Vol PAS-101 No. 6 pp. 1257-

1270 June 1982.

0.00551 0.00750 0.003 13

0.00563 0.00887 0.00325

THREE 0.00644 0.01009 0.00367

0.00620 0.00975 0.00356

Table 6.a. 3rd harmonic voltages for case E.

[2] W. Grady, "Harmonic Power Flow Studies". PhD Thesis,

Purdue University, West Lafayette, IN. August, 1983.

[3] EPRI EL-3300, "Harmonic Power FLOW tud es Volume

1: Theoretical Basis". Project 1764-7,FinalReport, No. 1983.

ms". PhD4] L. Kraft, "Harmonic Resonance in Power Svste

Thesis, Purdue University, West Lafayette, IN,August 1984.

[5] G. Heydt and W. rady, "Distributed Rectifier Loads in

Electric Power Sytems". JEEE Trans.. PAS -103. NQ6,PP1385-1390. June 1985.



2058

[6] J.G. Mayordomo, "&idvs s of the Harm0nics Invected irrETS de II. Madrid, 1986.(In Spanish).fie NetWOrks due to Powe Converters". PhD Dissertation.

[7] J.G. Mayordomo, A. Perez Coyto,"ComputerProgram forAnalyzing Converter Harmonics in Power Systems.Application to Static VAR Compensation Analysis." Proc.Q€

, p. 115-124. Madrid, September 1987.

IEEE International Workshop on Control Svs&ms in New

prier-

[8] M. Valckcel and J.G. Mayordomo, "A SimplifiedHarmonic Power Flow". J"c. ASTED Power H eh Tech'89,

pp 241-246. Valencia (Spain), July 1989.

[9 ] M. Valchcel. "&ady -State Analvsis of Electrical Power

Svstems with Non-linear Elements bv means o a n Harmonic

Power Flow Method".PhD Dissertation. ETS de 11. Madrid,

1991. (In Spanish).

[ lo] H.W. Dommel,*'Electromaenet'C Transients Proerm

Reference Manual lEMTPTheoryBook)." Bonneville PowerAdministration, Portland, Oregon. August 1986.

[ l l ] D. Pileggi, N. Chandra and A. Emanuel, "Prediction ofHarmonic Voltage in Distribution Systems". IEEE Trans,

. .

Ees.Vol PAS-101, pp 1307-1315,March 1981.

[123 M. Depenbrock, "Wirk-und Blindleinstungen PeriodischerStr6me in Ein und Mehrphasensystemen mit periodischenSpannungen beliebiger kurvenform". ETG Fachbe ichte Uber:

e. VDE-Verlu, pp. 17-62, 1979.

[13] J. Arrillaga and C.D. Callaghan, "ThreePhase AC-DCLoad and Harmonic Flows". JEEE Trans on Power Del very.

Vol. 6, NP1 pp 238-244, Jan~ary 991.

[14] W. Xu, J.R. Marti and H.W. Dommel, "A MultiphaseHarmonic Load Flow Solution Technique". IEEE PES Winter

Meeting, NQ 0 WM 098-4 PWRS, Atlanta, Georgia, February

1990.

[15] W. Xu, J.R. Marti and H.W. Dommel, "Harmonic

Analysis Systems with Static Compensators". IEEE PES

Winter Meeting, NQ 0 WM 99-2 PWRS, Atlanta, Georgia,

February 1990.

[16] J.P. Tamby and VJ. John, "Q'Harm- A Harmonic PowerFlow Program for Small Power Systems". IEEE Trans. onPower Svstems. Vol. 3, NQ3, p 949-955. August 1988.

APPEND X

THREE-PHASE C O W TIONAL LOADS

1 . STRUCTLTRALLY BALANCED LOAD,

The current in phase a,I, is:

I , = Y eY U , = Y , ( U 1 + U 2 ) (A. 1)

P, - j ' Q , - P, - j Q,Y = 2 2 2 - 2 ( A . 4 )

u , + u + U , 9 ( U 1 + U 2 )bc

The expressions for the sequence currentsare:

2. STRUCIURALLYUNBALANCED OAD,

The threebranches are differentin this type of load. Then, it is

necessaryto schedule activeand reactive powers in each branch

Qab, P b c s Qbc*P c ~nd Q c ~

In this figure,

1, = u & Y & - U , Y ,

I , = U Y

I c = U C r Y , - U , YbC

bc b ~ - ~ a b ~ a b

The three branches are identical in this typeof load. Only total

active and reactive powers (Pt andQt)are scheduled.



2059

Where:* 2

ab = ab' 'ab

* 2

* 2

Y , = S , / U b C

Y , = S c P / U C .

2

U a b = U a - U b = ( l - a ) U 1 + ( 1 - a ) U 2

ubC u b - U = = ( a 2 - a ) u 1+ ( a - a 3 U , (A . 8 )

U, = u C- U, = ( a - 1) u1+ (a' - 1 ) U,

U a b = 3 [ U l + u 2 + 2 U I u 2 c 0 s2 2 ( 9 , - 9 , +a0]

Sequence currents (expression (A.11)) aredefined in terms of

sequence voltages and scheduled phase powers by combining

(A.9) and (A.12).

U , = 3 [ U 1 + U 2 + 2 U 1 U 2 c 0 s2 2 ( e l - + 2 -180°] (A .9)

U,2 = 3 [ U l + U 2 + 2 U l U 2 c 0 s2 ( 9 , - 9 , -60'1

The phase currents are obtained by introduction of (A.7) and(A.8) into (A.6). The sequence components are obtained by

means of the general transformation:

I = 1 ( 1 ~ + a 1 a I ~ )

I = l ( ~ ~ + aI ~ + I ~ )

1 3 b

(A . 10)

2 3

Combining the previous expressions, sequence currents can bewrite in the next form:

I 1 ( A + j B ) U 1 + ( C j D) ,

1 2 = ( E + F ) U 1 + ( K + j L ) U 2

(A . 11)

where:2 2 2

ab bc= K = P a b / U + P / U , + P , / U ,

B = L = - Q a b / U a b - Q b C / U L - Q , / U , 2

2 2C=(P,- Q, / 3 ) / ( 2 U a b ) - Pbc U, +

was born in Silleda (Ponteveb), Spain. He

received his B.S. degree and his PhD degree in Electrical

Engineering from E.T.S. de Ingenieros Industriales(Polytechnic University of Madrid) in 1982 and 1991respectively. In 1983 he joined the LaboratorioCentral Oficial

de Electrotecnia (LCOE) in the E.T.S. de IngenierosIndustrialesas Responsible of the Network Analysis Section.

His research interests include steady-state power systems andharmonic analysis in power systems.

Julio G. was born in Madrid, Spain. He received

his B S.degree and his PhD degree in Electrical Engineeringfrom E.T.S. de Ingenieros Industriales PolytechnicUniversity

of Madrid) in 1980and 1986. In 1980 he joinedtheDepartment

of Electrical Engineering where he is presently Lecturer of

Electrical Engineering. His research interests include transientphenomena in networks and harmonic analysis in power

systems.

Documents

Harmonic Power Flow for Unbalanced Systems