–259–

I. Introduction

Distribution load flow is a very important tool forthe analysis of distribution systems and is used in opera-tional as well as planning environments as described byLin and Chen (1986), the IEEE Tutorial Course on Distri-bution Automation (IEEE, 1988) and the IEEE Tutori-al Course on Power Distribution Planning (IEEE, 1992).Many real-time applications in the distribution automa-tion system (DAS) and distribution management system(DMS), such as network optimization, Var planning,switching, state estimation and so forth, need the supportof a robust and efficient power flow method. Such apower flow solution must be able to model the special fea-tures of distribution systems in sufficient detail. Some ofthe inherent features of electric distribution systems are:

(1) a radial network structure,(2) an unbalanced distributed load and unbalanced op-

eration,(3) an extremely large number of branches/nodes, and(4) a wide range of resistance and reactance values.

These features cause the traditional power flow methods,the Gauss-Seidel and Newton-Raphson techniques, whicharise from the transmission area, to lack either computereconomy or robustness in distribution applications. Inparticular, the assumptions necessary for the simplifi-cations used in the standard fast-decoupled Newton-Raphson method as reported by Stott and Alsac (1974) are

often not valid for distribution systems. In addition to per-formance, all the above mentioned characteristics need tobe considered to qualify a good distribution load flow al-gorithm.

Some efforts have been made to overcome these dif-ficulties. Some of these methods, such as those of Birt etal. (1976), Chen et al. (1991a, 1991b), Chen and Chang(1992), and Teng and Lin (1994), based on the generalmeshed topology of a transmission system, are also appli-cable to distribution systems. The Gauss implicit Z-ma-trix method as reported by Chen et al. (1991a, 1991b)does not explicitly exploit the radial structure of the distri-bution system and, therefore, requires finding the solutionof a set of equations whose size follows the order of thenumber of buses. Recent researches have led to some newideas on how to deal with distribution networks. Theseideas may require new data formats, such as the compen-sation-based technique reported by Shirmohammadi et al.(1988), where the forward-backward sweep algorithm isadopted in the solution scheme. Luo and Semlyen (1990)requires a definition for feeder breakpoints, which has tobe specified by examining the feeder segments one-by-one. Cheng and Shirmohammadi (1994) adopts the feed-er-lateral based model, which requires the “layer-lateral”based data format. In addition to the conventional bus-branch oriented data format, new data bases have to bebuilt and maintained for these models to run.

The algorithm proposed in this paper is a “novel but

Proc. Natl. Sci. Counc. ROC(A)Vol. 24, No. 4, 2000. pp. 259-264

A Network-Topology-based Three-Phase Load Flow for

Distribution Systems

JEN-HAO TENG

Department of Electrical EngineeringI-Shou University

Kaohsiung, Taiwan, R.O.C.

(Received August 16, 1999; Accepted December 17, 1999)

ABSTRACT

A network-topology-based three-phase distribution power flow algorithm is developed in this paper. Thespecial topology of a distribution network has been fully exploited to make obtaining a direct solution possible.Two developed matrices are enough to obtain the power flow solution: they are the bus-injection to branch-currentmatrix and the branch-current to bus-voltage matrix. The traditional Newton Raphson and Gauss implicit Z matrixalgorithms, which need LU decomposition and forward/backward substitution of the Jacobian matrix or the Yadmittance matrix, are not needed for this new development. The features of this method are robustness and com-puter economy. Tests show that the proposed method converges in almost all circumstances for distribution sys-tems and has great potential for use with distribution automation system.

Key Words: distribution automation system, distribution load flow, distribution management system, Gaussimplicit Z matrix method, Newton-Raphson method

classic” technique, running on the conventional bus-branch oriented data format used by most utilities. Ourgoal is to develop a formulation which exploits the topo-logical characteristics of a distribution system and solvesthe distribution load flow directly. This means that thetraditional Newton Raphson and Gauss implicit Z matrixalgorithms, which need LU decomposition and forward/backward substitution of the Jacobian matrix or the Yadmittance matrix, are not needed in this new develop-ment. Two developed matrices, the bus-injection tobranch-current matrix and branch-current to bus-voltagematrix, and a simple matrix multiplication are utilized toobtain the power flow solution. The features of this me-thod are robustness and computer economy. Tests showthat the proposed method has great potential for real-timeoperation.

II. Unbalanced Three-Phase Model

Figure 1 shows a three-phase line section betweenBus i and j. The line parameters can be obtained using themethod developed by Carson (1926). A 4 × 4 matrix,which takes into account the self and mutual couplingterms, can be expressed as

(1)

For a well-grounded distribution system, VN and Vn shownin Fig. 1 are assumed to be zero, and Kron’s reduction canbe applied in Eq. (1). Equation (2) is designed to includethe effects of the neutral or ground wire and to be used inthe unbalanced load flow calculation:

(2)

The relations between the bus voltages and branch cur-rents in Fig. 1 can be expressed as

(3)

For any phase which fails to present, the correspond-ing row and column in this matrix will contain null-en-tries.

III. Formulation Development

The proposed method is based on two matrices, thebus-injection to branch-current matrix and branch-currentto bus-voltage matrix, and on the equivalent current injec-tion. They are developed in this section.

1. Equivalent Current Injection

For distribution systems, the models which are basedon the equivalent current injection, as reported by Shir-mohammadi et al. (1988), Chen et al. (1991a) and Tengand Lin (1994), are more convenient to use. At each Busi, the complex power Si is specified by

(4)

and the corresponding equivalent current injection at thek-th iteration of the solution is

(5)

where

Vki is the node voltage at the k-th iteration;

Iki is the equivalent current injection at the k-th itera-

tion;

Iri and Ii

i are the real and imaginary parts of the equiv-alent current injection at the k-th iteration, respec-tively.

2. Building Algorithms for Developed Matrices

A. Bus-Injection to Branch-Current Matrix

The simple distribution system shown in Fig. 2 willbe used as an example. The power injections can be con-

I I V jI VP jQ

Vik

ir

ik

ii

ik i i

ik= + = +

( ) ( ) ,*

S P jQ i Ni i i= + =( ) , , , , 1 2 L

V

V

V

V

V

V

Z Z Z

Z Z Z

Z Z Z

I

I

I

a

b

c

A

B

C

aa n ab n ac n

ba n bb n bc n

ca n cb n cc n

Aa

Bb

Cc

=

−

− − −

− − −

− − −

.

Zabc

aa n ab n ac n

ba n bb n bc n

ca n cb n cc n

Z Z Z

Z Z Z

Z Z Z

[ ] =

− − −

− − −

− − −

.

Zabcn

aa ab ac an

ba bb bc bn

ca cb cc cn

na nb nc nn

Z Z Z Z

Z Z Z Z

Z Z Z Z

Z Z Z Z

[ ] =

.

J.H. Teng

–260–

Fig. 1. A three-phase line section.

verted into the equivalent current injections using Eq. (5),and a set of equations can be written by applying Kir-chhoff’s Current Law (KCL) to the distribution network.Then, the branch currents can be formulated as a functionof the equivalent current injections. For example, thebranch currents B5, B3 and B1 can be expressed as

(6)

Furthermore, the Bus-Injection to Branch-Current (BIBC)matrix can be obtained as

(7a)

Equation (7a) can be expressed in the general form as

(7b)

The constant BIBC matrix has non-zero entries of +1only.

By inspecting Eq. (7), we can develop a buildingalgorithm for the BIBC matrix as follows:

Procedure (1) – For a distribution system with m-branch sections and an n-bus, the di-mension of the BIBC matrix is m × (n– 1).

Procedure (2) – If a line section (Bk) is located be-tween Bus i and Bus j, copy the col-

umn of the i-th bus of the BIBC ma-trix to the column of the j-th bus andfill a + 1 in the position of the k-th rowand the j-th bus column.

Procedure (3) – Repeat Procedure (2) until all the linesections are included in the BIBC ma-trix.

The building Procedure (2) for the BIBC matrix is shownin Fig. 3.

The algorithm can be easily expanded to a multi-phase line section or bus. For example, if the line sectionbetween Bus i and Bus j is a three-phase line section, thenthe corresponding branch current Bi will be a 3 × 1 vector,and the +1 in the BIBC matrix will become a 3 × 3 identi-ty matrix.

B. Branch-Current to Bus-Voltage Matrix

The relations between the branch currents and busvoltages as shown in Fig. 2 can be obtained by using Eq.(3). For example, the voltages of Bus 2, 3, and 4 are

(8a)

(8b)

(8c)

where Vi is the bus voltage of Bus i, and Zij is the lineimpedance between Bus i and Bus j.

Substituting Eqs. (8a) and (8b) into Eq. (8c), thevoltage of Bus 4 can be rewritten as

(9)

From Eq. (9), it can be seen that the bus voltage can beexpressed as a function of the branch currents, line param-eters and substation voltage. Similar procedures can beutilized for other buses, and the Branch-Current to Bus-Voltage (BCBV) matrix can be derived as

(10a)

Rewriting Eq. (10a) in the general form, we have

(10b)∆[ ] = [ ][ ]V BCBV B .

V

V

V

V

V

V

V

V

V

V

Z

Z Z

Z Z Z

Z Z Z Z

Z Z Z

1

1

1

1

1

2

3

4

5

6

12

12 23

12 23 34

12 23 34 45

12 23 36

0 0 0 0

0 0 0

0 0

0

0 0

−

=

B

B

B

B

B

1

2

3

4

5

.

V V B Z B Z B Z4 1 1 12 2 23 3 34= − − − .

V V B Z4 3 3 34= − ,

V V B Z3 2 2 23= − ,

V V B Z2 1 1 12= − ,

B BIBC I[ ] = [ ][ ] .

B

B

B

B

B

I

I

I

I

I

1

2

3

4

5

2

3

4

5

6

1 1 1 1 1

0 1 1 1 1

0 0 1 1 0

0 0 0 1 0

0 0 0 0 1

=

.

B I

B I I

B I I I I I

5 6

3 4 5

1 2 3 4 5 6

=

= +

= + + + +

,

,

.

A Rapid Distribution Load Flow

–261–

Fig. 2. A simple distribution system.

Based on Eq. (10), a building algorithm for the BCBVmatrix can be developed as follows:

Procedure (4) – For a distribution system with m-branch sections and an n-bus, the di-mension of the BCBV matrix is (n – 1)× m.

Procedure (5) – If a line section (Bk) is located be-tween Bus i and Bus j, copy the row ofthe i-th bus of the BCBV matrix to therow of the j-th bus, and fill the lineimpedance (Zij) in the position of the j-th bus row and the k-th column.

Procedure (6) – Repeat Procedure (5) until all the linesections are included in the BCBV ma-trix.

The building Procedure (5) for the BCBV matrix is shownin Fig. 4.

The algorithm can be expanded to a multi-phase linesection or bus easily. For example, if the line section be-tween Bus i and Bus j is a three-phase line section, thenthe corresponding branch current Bi will be a 3 × 1 vector,and Zij in the BCBV matrix will be a 3 × 3 impedancematrix as shown in Eq. (2).

From Figs. 3 and 4, it can be seen that the buildingalgorithms for the BIBC and BCBV matrices are similar.In fact, these two matrices were built in the same subrou-tine of our test program. Therefore, the amount of compu-tation resources needed can be reduced. In addition, thebuilding algorithms are based on the traditional bus-branch oriented data base, so the data preparation time ofthe proposed algorithm can be reduced and can be inte-grated into the existing DAS.

3. Solution Techniques

The BIBC and BCBV matrices were developedbased on the topological structure of distribution systems.The BIBC matrix is responsible for the relations betweenthe bus current injections and branch currents. The corre-

sponding variation of the branch currents, which is gener-ated by the variation at the current injection buses, can befound directly by using the BIBC matrix. The BCBVmatrix is responsible for the relations between the branchcurrents and bus voltages. The corresponding variation ofthe bus voltages, which is generated by the variation of thebranch currents, can be found directly by using the BCBVmatrix. Combining Eqs. (7b) and (10b), the relations be-tween the bus current injections and bus voltages can be ex-pressed as

(11)

and the solution for the distribution load flow can beobtained by solving Eqs. (12a) and (12b) iteratively:

(12a)

(12b)

Compared with the traditional Newton Raphson andGauss implicit Z matrix algorithms, which need LU de-composition and forward/backward substitution of theJacobian matrix or the Y admittance matrix, the new for-mulation uses only the DLF matrix to solve load flowproblem. The time-consuming LU decomposition and for-ward/backward substitution procedures are not needed.This considerably reduces the amount of computation re-sources needed and makes the proposed method suitablefor on-line operation.

The proposed algorithm is summarized as follows:(1) Input data.(2) Use Procedures (1), (2), (3) and Eq. (7) to form the

BIBC matrix.(3) Use Procedures (4), (5), (6) and Eq. (10) to form

the BCBV matrix.

∆[ ] = [ ][ ]+V DLF Ik k1 .

I I V jI VP jQ

Vik

ir

ik

ii

ik i i

ik= + = +

( ) ( ) ( ) ,*

∆[ ] = [ ][ ][ ]

= [ ][ ]

V BCBV BIBC I

DLF I ,

J.H. Teng

–262–

Fig. 3. The building Procedure (2) for the BIBC matrix.

Fig. 4. The building Procedure (5) for the BCBV matrix.

(4) Use Eq. (11) to form the DLF matrix.(5) Iteration k = 0.(6) Iteration k = k + 1.(7) Solve for the three-phase power flow by using Eqs.

(12a) and (12b), and update voltages.(8) If max

i ( Ik+1i – Ik

i ) > tolerance, goto (6).(9) Report and end.

IV. Test Results

The proposed three-phase power flow program wasimplemented using the Borland C++ language and testedon a Windows-98 based Pentium-II (350) PC. Two meth-ods were used in the tests, and the convergence tolerancewas set at 0.001.

Method 1: The Gauss implicit Z-Bus method as report-ed by Chen et al. (1991a).

Method 2: The proposed algorithm.

1. Accuracy Comparison

For any new method, it is important to make surethat the final solution obtained using the proposed methodis the same as that obtained using the existing method. Asimple 8-bus system (equivalent 13-bus system), includingthree-phase, double-phase and single-phase line sectionsand buses, is shown in Fig. 5. The final voltage solutionsobtained using Method 1 and Method 2 are shown inTable 1. From Table 1, it can be seen that the final con-verged voltage solutions obtained using Method 1 are veryclose to the solutions obtained using Method 2. Thismeans that the proposed method can be used to solve forthe distribution load flow.

2. Performance Tests

The test feeders were 13, 37 and 123 bus, three-phase IEEE test feeders as reported by Kersting (1991).The feeders were predominantly three-phase lateral with

unbalanced loads. The execution time and number foriterations for these two methods are shown in Table 2.From Table 2, it can be seen that Method 2 outperformedMethod 1, especially in the case of a large-scale distribu-tion system since time-consuming procedures, such as LUdecomposition and forward/backward substitution, are notneeded in Method 2. Moreover, the results shown inTable 2 reveal that the number of iterations needed byMethod 2 is stable. Method 2 is, definitely, a robust algo-rithm.

V. Discussion and Conclusion

In this paper, a direct approach algorithm for distri-bution load flow has been developed. The features of thismethod are robustness and computer economy. Two ma-trices, developed based on the topological structure of dis-tribution systems, have been used to solve the load flowproblem. The BIBC matrix is responsible for the varia-tion between the bus current injection and branch current,and the BCBV matrix is responsible for the variationbetween the branch current and bus voltage. The pro-posed solution algorithm is primarily based on these twomatrices and matrix multiplication. Time-consuming pro-cedures, such as LU factorization and forward/backwardsubstitution of the Jacobian matrix are not needed, and the

A Rapid Distribution Load Flow

–263–

Fig. 5. An 8-bus distribution system.

Table 1. Final Converged Voltage Solutions

BusMethod 1 Method 2

Number |V|(pu) Angle |V|(pu) Angle Phase(Rad.) (Rad.).

1 1.0000 0.0000 1.0000 0.0000 A1 1.0000 –2.0944 1.0000 –2.0944 B1 1.0000 2.0944 1.0000 2.0944 C2 0.9840 0.0032 0.9839 0.0032 A2 0.9714 –2.0902 0.9712 –2.0902 B2 0.9699 2.0939 0.9697 2.0939 C3 0.9833 0.0031 0.9832 0.0031 A4 0.9653 –2.0897 0.9652 –2.0897 B4 0.9672 2.0932 0.9669 2.0932 C5 0.9644 –2.0898 0.9640 –2.0898 B6 0.9652 2.0930 0.9650 2.0930 C7 0.9686 2.0937 0.9683 2.0937 C8 0.9674 2.0936 0.9671 2.0936 C

Table 2. Execution Time and Number of Iterations

Method 1 Method 2

Feeder Execution Number of Execution Number ofTime (sec.) Iterations Time (sec.) Iterations

IEEE-13 0.0165 3 0.00565 3IEEE-37 0.2190 3 0.0181 3IEEE-123 2.4453 4 0.1127 4

ill-conditioned problem which occurs at the Jacobian ma-trix does not exist in the solution procedure. Therefore,the proposed method is robust and economical. Testresults show that the proposed method is suitable forpower flow calculations in large-scale distribution sys-tems. Other issues involved in distribution system opera-tion, such as multi-phase operation with unbalanced anddistributed loads, voltage regulators and capacitors withautomatic tap controls, will be discussed in a future paper.

Acknowledgment

This paper was sponsored by the National Science Council,R.O.C., under research grant NSC 88-2213-E-214-041. The authorwould like to thank Dr. Shun-Yu Chan for his useful comments on thispaper. The author would also like to thank the reviewers for their contri-butions to this paper.

References

Carson, J. R. (1926) Wave propagation in overhead wires with groundreturn. Bell System Technical Journal, 5, 1-7.

Birt, K. A., J. J. Graffy, and J. D. McDonald (1976) Three phase loadflow program. IEEE Trans. on Power Apparatus and Systems, PAS-95, 59-65.

Chen, T. H. and J. D. Chang (1992) Open wye-open delta and opendelta-open delta transformer models for rigorous distribution systemanalysis. IEE Proceedings-C, 139, 227-234.

Chen, T. H., M. S. Chen, T. Inoue, P. Kotas, and E. A. Chebli (1991a)Three-phase co-generator and transformer models for distributionsystem analysis. IEEE Trans. on Power Delivery, 6, 1671-1681.

Chen, T. H., M. S. Chen, K. J. Hwang, P. Kotas, and E. A. Chebli(1991b) Distribution system power flow analysis a rigid approach.IEEE Trans. on Power Delivery, 6, 1146-1152.

Cheng, C. S. and D. Shirmohammadi (1994) A three-phase power flowmethod for real-time distribution system analysis. IEEE SummerMeeting, SM. 603-1 PWRS, San Francisco, CA, U.S.A.

IEEE (1988) IEEE Tutorial Course on Distribution Automation. 88 EHO280-8 PWR, IEEE Power Engineering Society, Piscataway, NJ,U.S.A.

IEEE (1992) IEEE Tutorial Course on Power Distribution Planning. 92EHO 381-6 PWR, IEEE Power Engineering Society, Piscataway,NJ, U.S.A.

Kersting, W. M. (1991) Radial distribution test feeders. IEEE Trans. onPower System, 6, 975-985.

Lin, W. M. and M. S. Chen (1986) An overall distribution automationstructure. Electric Power Systems Research, 10, 7-19.

Luo, G. X. and A. Semlyen (1990) Efficient load flow for large weaklymeshed networks. IEEE Trans. on Power Systems, 5, 1309-1316.

Shirmohammadi, D., H. W. Hong, A. Semlyen, and G. X. Luo (1988) Acompensation-based power flow method for weakly meshed distri-bution and transmission networks. IEEE Trans. on Power Systems,3, 753-762.

Stott, B. and O. Alsac (1974) Fast decoupled load flow. IEEE Trans. onPower Apparatus and Systems, PAS-93, 859-867.

Teng, J. H. and W. M. Lin (1994) Current-based power flow solutions fordistribution systems. IEEE ICPST, pp. 414-428, Beijing, P.R.C.

Zimmerman, R. D. and H. D. Chiang (1995) Fast decoupled power flowfor unbalanced radial distribution systems. IEEE Winter Meeting,WM. 219-6 PWRS, New York, NY, U.S.A.

J.H. Teng

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- Z LU Y