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

Distribution Systems

By:-Gaurav Ranjan (06P61A0220)

ABSTRACT:

A network-topology-based three-phase distribution power flow algorithm is developed in this paper.

The special topology of a distribution network has been fully exploited to

obtain a direct solution.Two developed matrices are enough to obtain the power flow

solution: they are the bus-injection to branch-current matrix and the branch-current to bus-voltage matrix.

The traditional Newton Raphson and Gauss implicit Z matrix algorithms, which need LU decomposition and forward/backward substitution of the Jacobian matrix or the Y admittance matrix, are not needed for this new development.

The proposed method is robust and has computer economy.Tests show that the proposed method converges in almost all

circumstances for distribution systems and has great potential for use with distribution automation system.

Introduction

Distribution load flow is a very important tool for the analysis of distribution systems and is used in operational as well as planning environments.

Many real-time applications in the distribution automation system (DAS) and distribution management system (DMS), such as network optimization, Var planning, switching, state estimation and so forth, need the support of a robust and efficient power flow method.

Some of the inherent features of electric distribution systems are:

a radial network structure an unbalanced distributed load and unbalanced operation, an extremely large number of branches/nodes, and a wide range of resistance and reactance values.

Unbalanced Three-Phase Model

The line parameters( 4 × 4 matrix) can be obtained using the method developed by Carson.

---(1)

For a well-grounded distribution system, VN and Vn are assumed to be zero, and Kron’s reduction can be applied.

---(2)

Equation (2) is designed to include the effects of the neutral or ground wire and to be used in the unbalanced load flow calculation:The relations between the bus voltages and branch currents in Fig. 1 can be expressed as

---(3)

FORMULATION DEVELOPMENTThe proposed method is based on two matrices, the bus-injection to branch-current matrix and branch-current to bus-voltage matrix, and on the equivalent current injection.

Equivalent Current InjectionAt each Bus i, the complex power Si is specified by

Si = (Pi + jQi ) i = 1, 2, L, N ---(4)

Equivalent current injection at the k-th iteration of the solution is

---(5)

Building Algorithms for Developed Matrices

A. Bus-Injection to Branch-Current Matrix

The power injections can be converted into the equivalent current injections using Eq. (5).

Applying Kirchhoff’s Current Law (KCL) to the distribution network, the branch currents can be formulated as a functionof the equivalent current injections.

B5=I6,B3=I4+I5 ---(6)B1=I2+I3+I4+I5+I6

The branch currents B5, B3 and B1 can be expressed as

The Bus-Injection to Branch-Current (BIBC) matrix can be obtained as

---(7a)

can be expressed in the general form as

[B]=[BIBC][I] ---(7b)

Algorithm for the BIBC matrix

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

Procedure (2) – If a line section (Bk) is located between Bus i and Bus j, copy the column of the i-th bus of the BIBC matrix to the column of the j-th bus and fill + 1 in the position of the k-th row and the j-th bus column as shown below.

Procedure (3) – Repeat Procedure (2) until all the line sections are included in the BIBC matrix.

B. Branch-Current to Bus-Voltage Matrix The relations between the branch currents and bus voltages as shown

in Fig. 2 can be obtained by using Eq. (3). For example, the voltages of Bus 2, 3, and 4 are

V2=V1-B1Z12 ----------(8a)

V3=V2-B2Z23 ----------(8b)

V4=V3-B3Z34 ----------(8c)

The voltage of Bus 4 can be rewritten as

V4= V1-B1Z12-B2Z23−B3Z34 ----------(9)

The bus voltage can be expressed as a function of the branch currents, line parameters and substation voltage

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

[ΔV] = [BCBV][B]. ----(10b)

---(10a)

Building algorithm for the BCBV matrix • Procedure (4) – For a distribution system with m branch sections and an

n-bus, the dimension of the BCBV matrix is (n – 1)× m.• Procedure (5) – If a line section (Bk) is located between Bus i and Bus j,

copy the row of the i-th bus of the BCBV matrix to the row of the j-th bus, and fill the line impedance (Zij) in the position of the jth bus row and the k-th column (as shown below).

• Procedure (6) – Repeat Procedure (5) until all the line sections are included in the BCBV matrix. The building Procedure (5) for the BCBV matrix is shown in Fig. 4.

• The BIBC and BCBV matrices were developed based on the topological structure of distribution systems.

• The BIBC matrix is responsible for the relations between the bus current injections and branch currents.

• The corresponding variation of the branch currents, which is generated by the variation at the current injection buses, can be found directly by using the BIBC matrix.

• The BCBV matrix is responsible for the relations between the branch currents and bus voltages.

• The corresponding variation of the bus voltages, which is generated by the variation of the branch currents, can be found directly by using the BCBV matrix.

Solution Techniques

Combining Eqs. (7b) and (10b), the relations between the bus current injections and bus voltages can be expressed as

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

Compared with the traditional Newton Raphson and Gauss implicit Z matrix algorithms, which need LU decomposition and forward/backward substitution of the Jacobian matrix or the Y admittance matrix, the new formulation uses only the DLF matrix to solve load flow problem.

This considerably reduces the amount of computation resources needed and makes the proposed method suitable for on-line operation.

Summary of proposed algorithm

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

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 maxi (|Ik+1i |– |Ik

i|) > tolerance, goto (6).

9. Report and end.

TEST RESULTS• The proposed three-phase power flow program was

implemented using the Borland C++ language and tested on a Windows-98 based Pentium-II (350) PC.

• Two methods were used in the tests, and the convergence tolerance was set at 0.001.

• Method 1: The Gauss implicit Z-Bus method.• Method 2: The proposed algorithm.• ACCURACY COMPARISON

For any new method, it is important to make sure that the final solution obtained using the proposed method is the same as that obtained using the existing method.

• A simple 8-bus system (equivalent 13-bus system), including three-phase, double-phase and single-phase line sections and buses, is shown in Fig. 5.

• The final voltage solutions obtained using Method 1 and Method 2 are shown in Table 1.

From Table 1, it can be seen that the final converged voltage solutions obtained using Method 1 are very close to the solutions obtained using Method 2.

• The test feeders were 13, 37 and 123 bus, three phase IEEE test feeders .

• The feeders were predominantly three-phase lateral with unbalanced loads.

• The execution time and number for iterations for these two methods are shown in Table 2

Performance Tests

The proposed method is robust and has computer economy.

Based on the topological structure of distribution systems ,two matrices(BIBC and BCBV) have been formed.

The BIBC matrix is responsible for the variation between the bus current injection and branch current.

The BCBV matrix is responsible for the variation between the branch current and bus voltage.

The proposed solution algorithm is primarily based on these two matrices and matrix multiplication.

Discussion and Conclusion

Discussion and Conclusion(cont..)

Time-consuming procedures, such as LU factorization and forward/backward substitution of the Jacobian matrix are not needed.

The ill-conditioned problem which occurs at the Jacobian matrix does not exist in the solution procedure.

Test results show that the proposed method is suitable for power flow calculations in large-scale distribution systems

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

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

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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.

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References

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