Improved Three-Phase Power-Flow Methods Using Sequence Components - MA Akher - KM Nor - AHA Rashid

7/31/2019 Improved Three-Phase Power-Flow Methods Using Sequence Components - MA Akher - KM Nor - AHA Rashid

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO.3, AUGUST 2005 1389

0885-8950/$20.00 2005 IEEE

Improved Three-Phase Power-Flow Methods

Using Sequence ComponentsMamdouh Abdel-Akher, Khalid Mohamed Nor, Senior Member, IEEE, and Abdul Halim Abdul Rashid

AbstractThe paper presents the formulation and the so-

lution of the three-phase power-flow problem using sequence

components. An improved sequence component transformer

model and a decoupled sequence line model were used. As a

result, the three sequence admittance matrices are decoupled with

three-phase power system features in sequence components. The

sequence power-flow algorithm has been formulated such that the

single-phase power-flow programs can be called as routines for

solving the positive sequence network. The results for the proposed

sequence power-flow methods are identical to those obtained from

three-phase power-flow programs developed in phase components.

The computational efficiency and convergence of the proposed

sequence three-phase power-flow methods show that they are as

fast and as robust as conventional the Newton

Raphson method.

Index TermsImproved sequence transformer model, sequence

decoupled line model, sequence power-flow methods, three-phase

power flow.

NOMENCLATURE

0, 1, 2 Suffix for sequence components., , Suffix for phase components., Suffix or prefix refers to busbar indices., Prefix for series and shunt elements of a

line.

, , , , , Variable and vector for active, reactive,and apparent power.

, , , , , Variable and vector for current, voltage,and angle.

, , , Submatrices of the NewtonRaphson

Jacobian matrix.

, Fast decoupled constant gradient ma-trices.

, Variable and matrix for admittance.

I. INTRODUCTION

OWER FLOW is an important tool in power system plan-ning and operational studies. The single-phase power-

flow

algorithms assume a balanced power system operation and a bal-anced power system model. There are many cases where thesystem unbalance cannot be ignored due to unbalanced loads,

Manuscript received July 29,2004; revised December 3,2004. This work wassupported in part by the University of Malaya, Kuala Lumpur, Malaysia, underan IRPA grant project. Paper no. TPWRS-00409-2004.

M.Abdel-Akher and K. M. Nor are with the Department of Electrical Engi-neering, University of Malaya, Kuala Lumpur 50603, Malaysia (e-mail: [email protected];[email protected]).

A. H. A. Rashid is with Institute of Mathematical Sciences, University ofMalaya, Kuala Lumpur 50603, Malaysia (e-mail: [email protected]).

Digital Object Identifier 10.1 109/TPWRS.2005.851933 untransposedtransmission lines, and a combination of balanced with

unbalanced networks in distribution systems. Therefore, athree-phase power-flow program that deals with the unbalancedpower system will be a useful analytical tool.

A variety of three-phase power-flow algorithms have beenstudied for solving unbalanced power systems. These al-gorithms can be categorized into two groups. The first groupsolves a general network structure such as the NewtonRaphsonmethod [1][5], the fast decoupled method [3], [6], the hybridmethod [7], and the bus admittance method [8]. The secondgroup considers primarily the radial structure of distributionnetworks such as the compensation-based method [9].

Unbalanced power systems can be modeled using phase com-ponents without simplifications [10]. However, the advantageof the application of sequence components is that the size ofthe problem is effectively reduced in comparison to the phasecomponents approach. In NewtonRaphson power flow, the size

of the problem is reduced from a ( ) Jacobian matrixto a ( ) Jacobian matrix for positive sequence powerflow and two ( ) admittance matrices for negative se-quence and zero sequence nodal voltage equations. Zhang [3]showed that the factorization time of an admittance matrix of

order ( ) is 70% more than the total factorization timeof three admittance matrices of order ( ). In addition, the

sequence networks, positive sequence, negative sequence, andzero sequence can be solved using parallel computations [3].

Based on sequence components, the three-phase power-flowproblem in [2] is decomposed into three separate subproblems.The line mutual coupling is included by putting the three sub-problems into an iterative scheme. The transformer phase shiftsare included in the solution process by transforming the (6 6)transformer admittance matrix in phase components to itscounterparts in sequence components. The (6 6) transformeradmittance matrix in sequence components is included in theovera l l ( ) admi t tance mat r ix .

The decoupled sequence components transmission linemodel was introduced in [3]. This line model allows the

overall ( ) sequence component admittance matrixto be decoupled into positive sequence, negative sequence, andzero sequence admittance matrices. The decouplingcompensation power-flow methods [3], [8] use theconventional sequence transformer model [10], where thetransformer phase shifts are difficult to be incorporated.

In comparison to phase components, implementation of thesequence components approach so far faced two problems toanalyze the unbalanced power system. The first is the fact thatthe coupling in the untransposed transmission line in sequence

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1390 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO.3, AUGUST 2005

components model still exists. Second, the phase shifts intro-duced by special transformer connections are difficult to be rep-resented [10]. On the other hand, in phase components, the cou-pling between lines and the phase shifts are included in the phasecomponents model.

All previous implementations of unbalanced three-phasepower flow based on sequence components used a single-phase

power flow that is modified from the traditional balancedpower-flow algorithms. Zhang, however, suggested the possi-bility of sequence components power flow to use the traditionalbalanced single-phase power flow for solving the positivesequence network [3].

In this paper, the sequence components transformer model[2] and the decoupled sequence components transmissionline model [3] are used together for developing an improvedsequence power-flow solution. As a result, the untransposedtransmission lines as well as the phase shifts introduced byspecial transformer connections are considered in the se-quence components power-flow solution process. In addition,the injected powers and currents due to loads and untrans-

posed transmission lines have been formulated such that thesingle-phase power-flow programs can be called as routines forsolving the positive sequence power flow.

The proposed sequence power-flow methods are tested withdifferent case studies and compared with the sequence-decou-pled power-flow methods [3], [8] and the hybrid method [7].The performance of the proposed methods is further examinedby comparison with a phase-coordinates NewtonRaphson pro-gram.

II. SEQUENCE COMPONENT POWER SYSTEM MODEL

A. Sequence Generator ModelFig. 1 shows the generator model for power-flow programs.

The model is represented with three uncoupled sequence cir-cuits [3], [4], [11]. The positive sequence reactance and EMFbehind it are not introduced; this actually presents the case inthe single-phase power-flow model.

If there is unbalance in a power system, current will flow inboth the negative sequence and zero sequence components ofthe generator model, resulting in three-phase unbalanced volt-ages at the generator busbar.

B. Sequence Transformer ModelThe transformer model is established by transforming the

overall transformer admittance matrix in phase components toits counterparts in sequence components as follows [2]:

(1)

where

Fig. 1. Sequence component model of generator.

The resulting (6 6) sequence admittance matrix is used forconstructing the sequence component transformer model that isused for building the decoupled sequence admittance matrices.The models for different transformer connections are summa-rized in Table I. The positive sequence and negative sequencenetworks of the - transformers, in Table I, are expressed byadmittance matrices as the phase shift cannot be suitably incor-porated in an equivalent circuit. In Table I, if bothand are not ignored, the transformerequivalent circuits should be modified to the typical

symmetrical component -equivalent model [10].

C. Sequence Transmission Line ModelWhen a transmission line is balanced or transposed, the ad-

mittance matrix in phase variables will be full and symmetrical.Hence, the transmission line can be represented by three uncou-pled sequence circuits as follows:

(2)

On the other hand, if any transmission line is unbalanced oruntransposed, the phase admittance matrix will also be full butsymmetrical in one diagonal axis. Therefore, the sequence ad-mittance matrix will be full and unsymmetrical as follows:

(3)The sequence coupled line model is characterized by weakmutual coupling, so the coupled line model in Fig. 2 can bedecoupled into three independent sequence circuits [3]. This canbe achieved by eliminating the off-diagonal elements in (3) byreplacing them with certain compensation current injections atboth ends of the unbalanced line.

The injected currents for an unbalanced line (see Fig. 2) con-nected between busbar and busbar due to the off-diagonal

(4)and


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elements in both the shunt and series admittance matrices canbe calculated as follows.

1) Off-diagonal current injection in the series admittancematrix:


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TABLE I

SEQUENCE COMPONENT TRANSFORMER MODEL

(6)

In the traditional balanced power-flow studies, the positivesequence current injection is transformed to positive sequencepower injection as follows:

(7)

Now, (3) can be rewritten similarly to (2) for transposed linesas follows:

(8)Fig. 3 shows the three decoupled sequence networks that rep-resent the coupled line model in Fig. 2. The mutual coupling is

included by the current and power injections at the line busbars.Equations (2) and (8) are used for constructing the decoupledsequence admittance matrices, whereas the coupling effect isconsidered by the current and power injections in (6) and (7),

respectively.

III. SEQUENCE POWER-FLOW METHODS

A. Busbars Specifications

The sequence power-flow methods use single-phasepower-flow specifications that include three types.

Slack Busbar: At the slack busbar, both the positive se-quence voltage magnitude and the angle are specified

Fig. 2. Sequence coupled line model.

2) Off-diagonal current injection in the shunt admittancematrix at any busbar, say, bus , is as follows:

(5)

The final sequence current injections at the line busbars are

at bus

at bus or or Fig. 3. Sequence decoupled line model.

(6)


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ABDEL-AKHER et al.: IMPROVED THREE-PHASE POWER-FLOW METHODS USING SEQUENCE COMPONENTS 1393

Voltage Controlled BusbarsPV: The PV busbar is a gener-ator busbar in which both the positive sequence voltage magni-tude and the total generated power are specified

(10)

Load Busbars: For balanced loads, the specified power for each phase is calculated from the total power demand at the load

busbarOn the other hand, the specified powers for unbalanced loads

are specified individually for each phase as

(12)The specified three-phase powers in (11) and (12) are used

for calculating the phase components injected currents. The in-je ct ed cur re nt of pha se ( , , or ) is given as fo ll ows :

(13)

The injected currents in the sequence networks due to loads are calculated by transforming the phase components injected currents in (13) to their counterparts in sequence components

Then, the load sequence specifications for the three sequencenetworks are calculated from the sequence currentinjections in(14). In the traditional balanced power-flowstudies, the current injection of the positive sequencenetwork is modified to powerinjection

(15)

The total sequence specifications can be evaluatedfrom (6),(7), and (15). Equations (6) and (7) present the coupling

effect in the untransposed transmission lines, whereas (15)presents theinjection due to the actual load in the network. Hence,the finalsequence specifications at a busbar are calculated asfollows:where refers to the total number of untransposedtransmis-

sion lines connected to busbar , and the summation in (16) and (17) gives the total injected currents and powers at busbar due to the untransposed transmission lines.

B. Positive Sequence Power Flow

Equations (9), (10), and (16) present the positive sequence power-flow specifications that are the same as those used by any single-phase power-flow algorithm. Both (9) and (10) present

the generator-specified values that are constant and need not be

updated during the sequence power-flow solution process. Thespecified values in (16) do not mean only the loads in the actualnetwork but also include the coupling effect. This means that there may be, at a certain busbar, no actual load, but there is a specified load in the positive sequence network. Equation (16) needs to be updated and supplied to the single-phase power-flowroutine for each iteration in the overall solution process. The calculated positive sequence power for busbars power system is given as

(18)

For PV busbars, only the active positive sequence power mis-match needs to be calculated in (19) using the specified genera-tion in (10).

The traditional NewtonRaphson and fast decoupled single-phase power-flow methods were chosen for solving the positivesequence power flow due to their well-established usage in the industry. Other methods, such as the current injection method [12], may also be used with appropriate adjustment. The basic

NewtonRaphson method is described by (20), while the fast

(14)

(16)

(17)

(20)

(21)

or or (11)

o r o r

(18)

The positive sequence mismatch at busbar is given as


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decoupled method is given by

C. Negative Sequence and Zero Sequence Nodal

Voltage Equations

The negative sequence and zero sequence voltages are calcu-lated by solving (22) and (23), respectively, using the specifiedcurrent injections in (17)

(22)

(23)


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Fig. 4. Sequence NewtonRaphson method.

D. Sparse Linear SolverIn the proposed implementation, the power system sparsity is

exploited by using SuperLU library routines [13]. SuperLU is ageneral-purpose library for the direct solution of large, sparse,and nonsymmetrical systems of linear equations.

E.Sequence Power-Flow AlgorithmFig. 4 shows the sequence power-flow solution algorithm.

The dashed block shows the NewtonRaphson single-phasepower-flow routine, which is completely decoupled from theoverall solution process. The NewtonRaphson algorithm canbe replaced by a fast decoupled algorithm, as done in thisstudy or by any other single-phase power-flow algorithm. Thesequence power-flow solution process starts with the following:

1) constructing the sequence admittance matrices ac-cording to the sequence power system models inSection II;

2) factorizing the negative sequence and zero sequence ad-mittance matrices using SuperLU library routines;

3) calculating the specified generation for positive se-quence power flow that is fixed and need not be updatedduring the solution process;

4) calculating the injected phase components currentsdue to the specified loads based on an initial set of

three- phase voltages using (13) (the initial three-phasevoltages are used in the first iteration only; in thefollowing iterations, the updated three-phase voltagesare used);

5) transforming the injected phase components currents ateach busbar to their counterparts in sequence compo-nents using (14);

6) combining the injected sequence components powersand currents of the specified loads and untransposedlines together for calculating the final sequence speci-fied values using (16) and (17);

7) solving (22) and (23) by using the specified negative se-quence and zero sequence currents to compute the neg-

ative sequence and zero sequence voltages. The positive


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sequence specified powers in (16) are supplied to the single-phase power-flow routine for calculating the pos-itive sequence voltages;

8) calculate the new set of three-phase voltages;9) go to step 4.

The process is repeated until a certain preset permissible tol-erance is reached. In Fig. 4, the convergence is measured using positive sequence power mismatch criterion. Other convergence criteria can be applied, such as positive sequence voltage mis-match or phase voltage mismatch.

IV. RESULTS AND DISCUSSION

,, and .

Case D) Configuration ABC-ABC, well-transposed lines.

Case E) Configuration ABC-ABC, untransposed lines.All case studies were run on a Pentium 4, 2.66-GHz CPU with

a 512-kB cache, 512 MB of RAM, and a MS Windows 2000 op-erating system. Case C was used for comparison between per-formance of the sequence power-flow and the phase-coordinatesNewtonRaphson power-flow program.

The sequence power-flow methods utilize SuperLU linear solver [13] and component technology as a program-ming methodology [14], whereas the phase-coordinates NewtonRaphson program is a commercial grade program based on object-oriented methods using a different sparse linear

solver. These differences may have some slight contribution in

the performance of the programs. However, the large difference in performance between the sequence components methods and the phase-coordinates NewtonRaphson program allowsreasonable conclusions to be drawn on the performance of the proposed methods.

A. Balanced Power System and Balanced Load

Cases A and D present the balanced system case study. Theresults are identical with those obtained from single-phase power-flow programs. The results differ only in the expected 30 phase shift accompanied to - transformer connections.

TABLE II

RESULTS OF CASE CPROPOSED METHODS

B. Unbalanced Power System and Balanced LoadCases B and E present the unbalanced power system and bal-

anced load case study. The three-phase busbar voltages, total generation, and total system losses are given in both Tables VIIand VIII in the Appendix for cases B and E, respectively. The results are identical with those given in [7, Tables 5 and 6]. Theslight difference in the three-phase busbar voltages at certain

busbars, such as phase a' at busbar 32' is due to the gener-ator model [7]. The positive sequence reactance and the EMF behind it are introduced in [7], while in this paper, the generator model in Fig. 1 is used.

C. Unbalanced Power System and Unbalanced LoadCase C presents the case of untransposed transmission system

and unbalanced load at busbar"14. The results in Table II aredifferent from those in Table III, where the sequence decouplingcompensation methods are used [3], [8]. This is because the se-quence power-flow methods are less sensitive to errors in the in-jected currents of the untransposed transmission lines, i.e., theprogram will converge but with wrong results. These errors arecorrected in this paper by the derivation of the current injections due to untransposed transmission lines. Then, these current in-jections are combined with the sequence current injections due to the actual loads in one set of sequence specifications, as wasdescribed by (16) and (17).

A sample of the power-flow results for case C is given in Table IV. The results include two busbars that have specified values; the first is the load busbar 14, and the second is thegenerator busbar 312. Theload constraint at busbar 14forphases a, b, and c is equal to

, and MVA, respectively, whereas at the

The sequence power-flow methodssequence NewtonRaphson and sequence fast decoupledare developed ac-cording to the algorithm shown in Fig. 4. These methods are coded using two single-phase power-flow programs, NewtonRaphson and fast decoupled [14]. These routines are

used for solving positive sequence network without affecting

their capability of solving balanced power systems.

The accuracy of the sequence power-flow methods are tested and compared with the decoupling compensation power-flowmethods [3], [8] and the hybrid method [7], whereas the perfor-mances are examined by comparison with a phase-coordinatesNewtonRaphson program. A 345-kV test system [7] is used forstudying the following cases.

Case A) Configuration ABC-CBA, well-transposed lines.Case B) Configuration ABC-CBA, untransposed lines. Case

C) Configuration ABC-CBA, untransposed lines and

unbalanced load at busbar14,


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TABLE III TABLE VRESULTS OF CASE CPOWER-FLOW METHODS [3], [8] CONVERGENCE CHARACTERISTIC OF THE PROPOSED METHODS


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total generation constraint.

TABLE IV

SAMPLE OF POWER-FLOW RESULTS OF CASE C

generator busbar 312, the total specified power constraint isequal to 250 MW. Table IV shows that the power delivered to the load at each phase satisfies the load constraint at busbar 14.

The total power leaves the generator busbar 312 is equal toMW, which also satisfies the

D. Convergence Characteristics

Table V shows the convergence characteristic of the sequencepower-flow methods for the studied cases AE. The sequencespecifications and the three-phase voltages are updated after cal-culating each sequence voltage (positivenegativezero). If thesequence specifications are updated only once every iteration step, as shown in Fig. 4, the number of iterations required forconvergence will increase, depending on the degree of the un-balance in the network.

The table shows, in all studied cases, that the sequence NewtonRaphson and sequence fast decoupled methods have a comparable number of iterations. The impact of the negative

sequence and zero sequence networks seem to have made the

convergence characteristics of the sequence fast decoupled similar to the sequence NewtonRaphson, unlike the case ofbalanced power-flow programs.

E. Time and Memory

Requirements

In comparing theperformance between the

proposed sequence power-flow and the phase-coordinatesNewtonRaphson method, Case C was run 100 times. Theaverage CPU time andmemory usage are reported in Table VI.

The fast execution time and the low memory requirements of

the sequence power-flow methods are expected since the execu-

tion time and memory requirements mainly depend on the size of the problem to be solved. The Jacobian matrix, negative se-quence, and zero sequence admittance matrices of the sequenceNewtonRaphson for busbars and branches power system i s a ( ) n o n z e r o e l e m e n t s i n s t e a d o f ( ) nonzero elements for the phase-coordinates Jacobian matrix. This is a savings in CPU memory of about 83%. This savings is about the same as shown in Table VI. The difference can be attributed to other memory requirements arising from the differ-

ences in the code implementation and the linear solver.

The algorithm execution time totally depends on the solution process. In the sequence NewtonRaphson, apart from the neg-ative sequence and zero sequence admittance matrices that arefactorized only once in the beginning of the iteration process, a() Jacobian matrix is refactorized and updated for each iteration. In phase-coordinates NewtonRaphson, a ( ) Jacobian matrix is refactorized and updated for every iteration step. This leads to a large savings in execution time, as shown in Table VI. The result in Table VI is consistent with the expe-

rience reported in [3], which was discussed earlier in Section I.

The sequence fast decoupled method, when compared with the sequence NewtonRaphson, is between 20%-30% faster.

TABLE VI

PERFORMANCE IN TERMS OF EXECUTION TIME AND CPU MEMORY


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TABLE VIII

RESULTS OF CASE E

Fig. 5. Performance of sequence power-flow methods for high R/X ratio.

TABLE VII

RESULTS OF CASEB

This is less savings compared with single-phase power flow, be-cause in three-phase power flow, the sequence fast decoupledand the sequence NewtonRaphson have to solve the same neg-

ative and zero sequence nodal voltage equations.

F. Effect of High Ratio

The effect of increasing the line ratio is studiedforboththeproposed methods and the phase-coordinates NewtonRaphsonmethod using the test case C. The ratio is increased untilthe divergence occurred, as shown in Fig. 5. The figure shows thatthe sequence NewtonRaphson has similar characteristics to thephase-coordinates NewtonRaphson method.

Although sequence fast decoupled takes fewer iterations thansequence NewtonRaphson for the normal ratio, as givenin Table V, the number of iterations increases when theratio is increased, as shown in Fig. 5. Similar observations werereported in [3], that the sequence fast decoupled is less sensi-tive for unbalanced loads but sensitive for ratio. Also, itis noted with some investigations that the characteristic of the

proposed sequence fast decoupled method is similar to the bal-


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anced fast decoupled algorithm for high ratio.

V. CONCLUSION

The paper has presented the formulation and the solutionof the three-phase power-flow problem using sequence compo-nents. The results of busbar voltages for different studied casesare the same as those obtained from a three-phase power-flowprograms developed in phase components.

The improved sequence component transformer model andthe decoupled sequence line model overcome the disadvantageof using sequence components and at the same time retainingthe advantage of dealing with matrices of smaller size. Thisadvantage leads to a large savings in time and memory whenthe proposed methods are compared with a phase-coordinatesNewtonRaphson method. In addition, the proposed sequenceNewtonRaphson method has similar convergence characteris-tics to the phase-coordinate NewtonRaphson method.

The power and current injections of the sequence networkshave been formulated such that single-phase power-flow pro-grams can be used in the three-phase power-flow algorithm. Thisallows balanced and unbalanced power-flow algorithms to be in-

tegrated in a single application.

APPENDIX

Tables VII and VIII show the results of Cases B and E, re-spectively.


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ACKNOWLEDGMENT

The authors gratefully acknowledge the assistance rendered by the Department of Electrical Engineering, University ofMalaya, in the work reported in this paper.

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[3] X. P. Zhang, Fast three phase load flow methods, IEEE Trans. PowerSyst., vol. 11, no. 3, pp. 15471553, Aug. 1996.

[4] B. C. Smith and J. Arrillaga, Improved three-phase load flow usingphase and sequence components, in Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 145, May 1998, pp. 245250.

[5] P. A. N. Garcia, J. L. R. Pereria, J. R. S. Cameiro, V. M. Da Costa, and N.Martins, Three-phase power flow calculation using the current injection method, IEEE Trans. Power Syst., vol. 15, no. 2, pp. 508514, May2000.

[6] J. Arrillaga and C. P. Arnold, Fast decoupled three phase load flow,Proc. Inst. Elect. Eng., vol. 125, no. 8, pp. 734740, Aug. 1978.

[7] B. K. Chen, M. S. Chen, R. R. Shoults, and C. C. Liang, Hybrid three phase load flow, in Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol.137, May 1990, pp. 177185.

[8] X. P. Zhang and H. Chen, Asymmetrical three phase load flow based on symmetrical component theory, in Proc. Inst. Elect. Eng., Gen.,Transm., Distrib., vol. 137, May 1994, pp. 248252.

[9] C. S. Cheng and D. Shirmohammadi, A three-phase power flow methodfor real-time distribution system analysis,IEEE Trans. Power Syst., vol.10, no. 2, pp. 671679, May 1995.

[10] J. Arrillaga and N. R. Watson, Computer modeling of electrical powersystems, in Book, 2nd ed. New York: Wiley, 2001, pp. 1113.

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Appl., vol. 20, no. 3, pp. 720755,1999.[14]K. M. Nor, H. Mokhlis, and T. A. Gani, Reusability techniques in

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Mamdouh Abdel-Akher was born in Qena, Egypt on October 9, 1974. He

received the B.Sc. degree with first-class honors andM.Sc. degrees in electrical engineering from AssuitUniversity, Assuit, Egypt, in 1997 and 2002,respectively. He is currently working toward thePh.D. degree in the Department of ElectricalEngineering, University of Malaya, Kuala Lumpur,Malaysia.

Since 1999, he has been with the Department ofElectrical Engineering, Aswan Faculty of Engi-

neering, South Valley University, Qena, Egypt, as a Research Engineer, and since 2002, as an

Assistant Lecturer. His current research interest is in power system analysis andsimulation.

Khalid Mohamed Nor (M81SM92) was born inSungai Pelong, Selangor, Malaysia. He received the B.Eng. degree with first-class honors from the Uni-versity of Liverpool, Liverpool, U.K. He received theM.Sc. degree in 1978 and the Ph.D. degree in 1981,both from the University of Manchester Institute ofScience and Technology, Manchester, U.K.

He joined the University of Malaya, Kuala

Lumpur, Malaysia, as a Lecturer in 1981. He is currently a Professor in the Department of ElectricalEngineering. His research interests are in the field ofelectrical power system simulation and powerquality.

Abdul Halim Abdul Rashid was born in KualaLipis, Pahang, Malaysia, on October 20, 1953. In 1976, he received the B.Sc. degree with first-classhonors from the University of Aston, Birmingham, U.K. He received the M.Sc. degree in 1979 and the Ph.D. degree in 1982 from the University of

Manchester, Manchester, U.K.In 1982, he joined the University of Malaya, Kuala

Lumpur, Malaysia, as a Lecturer. He is currently anAssociate Professor in the Institute of MathematicalSciences. His research interests are in the field of the

numerical computation and optimization and simulation.

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Improved Three-Phase Power-Flow Methods Using Sequence Components - MA Akher - KM Nor - AHA Rashid