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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
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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).
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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
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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.
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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
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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.
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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.
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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.
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