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Power System Analysis and Computational method Group Assignment Load Flow Study Lecturer: Dr. Saifulnizam Abd.Khlid

Prepared By: Mehrdad Mokhtari & Mehran Khosravifard

September 2012

Formation of Ybus by Singular Transformation

Formation of

by Singular Transformation

Mehrdad Mokhtari, Mehran Khosravifard

I.

Introduction

Network modelsfor load flow studies can be formed by

and

. One of the effectual

alternative approaches to assemble

is use of singular transformation based on the graph

theory. In this regard, developing

by the concept of singular transformation based on

building block as primitive network and its corresponding matrix is defined.

II.

Primitive Network Matrices

A network is said to be primitive when the network elements are not interconnected with other part of the whole network. The primitive network can be modelled in impedance and admittance form. Therefore, the performance equations of such a network are defined for each model. Figures 1 and 2 illustrate the primitive branches in impedance and admittance form respectively with neglecting the mutual coupling between branches.

Figure 1: Primitive branch in impedance form

Figure 2: Primitive branch in admittance form

1

Formation of Ybus by Singular Transformation

Equation (1) and (2) describe the mentioned performance equations for impedance and admittance forms of network. V+E =zI

(1)

I+i=yV

(2)

Where, V and I are the branch voltage and current vectors, while E and i are source vectors.zand y refer to primitive impedance and admittance matrices, which have not any information regarding connection of elements.

III.

Network Graph Formation

A typical 3-bus, 3-line network shown in figure 4 is used to define the corresponding graph.

Figure 4: A three-bus three-line power system

Buses are shown as node and each impedance is designated as branch and shown as connected line between relevant buses. For the buses with generator, a reference node is assigned. Therefore, the oriented graph of the system can be shown in figure5.

2

Formation of Ybus by Singular Transformation b2 n2

n1 b1

b4 n0

b3 n3

Figure 5: Oriented graph of the system It is worth remarking that direction of currents are arbitrary because by transpose multiplication, the desired admittance elements of final admittance matrix will be achieved.

IV.

Incidence Matrix Formation

Owning to offer the connection information of branches in network, a connection matrix with the size of (b×n), which is termed as node incidence matrix [A] will be formed to supplement primitive matrices z and y. As convention of circuit theory, the branch orientations are defined as bellow:

= +1 when the current in the branch i away from node j. = -1 when the current in the branch i enters node j. = 0 when branch i is not connected to branch j.

Therefore designated incidence matrix [A] by considering oriented graph of the system in figure5, can be formed as bellow; b 1 A= 2 3 4

n= 1

2

3

3

4

Formation of Ybus by Singular Transformation

V.

Bus Admittance Matrix Formation

Through figure 2, (I+i) = yV, for branch b,

Pre-multiplying by

, transpose of the reduced incidence matrix, which is deprived of reference

node,

However, as the algebraic sum of the currents entering each bus is zero,

This gives or, Again,

is a vector sum of all source currents entering each bus and can be designated as

with the size of (n×1).

which

is bus voltage and can be written in, [I] = [

Finally, bus admittance matrix [

][V]

] with the size of (n×n) is achieved, [

Therefore, bus admittance matrix [

] of the system depicted in figure 4 can be formed as

bellow. First the node incident matrix of the system could be formed as matrix A:

4

Formation of Ybus by Singular Transformation

A=

The matrix would be smaller by deleting the column relates to referenced bus, and then the reduced indecent matrix would be formed:

A=

Now primitive matrix impedance would be:

[Z] =

And primitive admittance matrix would be formed by inversing the impedance matrix as below:

=[ ] =

[ ] . [A] =

=

And finally admittance matrix would form:

=

=

5

Formation of Ybus by Singular Transformation

Appendix A Admittance Matrix formation of a Given Simple Power System

G1

G2

j1

j0.8

j0.4

1

2

j0.2

j0.2 3 j0.08 4

Step1: Corresponding Graph Formation of Proposed Power System b3

n1

n2 b2

b1 b5

b4 n0

n3 b6 n4

Step2: Incidence Matrix Formation

A=

6

n0

Formation of Ybus by Singular Transformation

and

Step3: Primitive Impedance and Admittance Matrices Formation

[Z] =

[Y] =

=

Step4: Admittance Matrix Formation

=

[ ][A] =

=

[

]=

=

7

Formation of Ybus by Singular Transformation

References [1] D.P.Kothari and I.J.Nagrath, Modern Power System Analysis , 3rd edition, McGraw-Hill, 2003 [2] A.Chakrabarti and S.Halder, Power System Analysis Operation and Control , 2nd edition, PHI, 2008 [3] H.Sadaat, Power System Analysis , 2nd edition, McGraw-Hill, 1999

8

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Prepared By: Mehrdad Mokhtari & Mehran Khosravifard

September 2012

Formation of Ybus by Singular Transformation

Formation of

by Singular Transformation

Mehrdad Mokhtari, Mehran Khosravifard

I.

Introduction

Network modelsfor load flow studies can be formed by

and

. One of the effectual

alternative approaches to assemble

is use of singular transformation based on the graph

theory. In this regard, developing

by the concept of singular transformation based on

building block as primitive network and its corresponding matrix is defined.

II.

Primitive Network Matrices

A network is said to be primitive when the network elements are not interconnected with other part of the whole network. The primitive network can be modelled in impedance and admittance form. Therefore, the performance equations of such a network are defined for each model. Figures 1 and 2 illustrate the primitive branches in impedance and admittance form respectively with neglecting the mutual coupling between branches.

Figure 1: Primitive branch in impedance form

Figure 2: Primitive branch in admittance form

1

Formation of Ybus by Singular Transformation

Equation (1) and (2) describe the mentioned performance equations for impedance and admittance forms of network. V+E =zI

(1)

I+i=yV

(2)

Where, V and I are the branch voltage and current vectors, while E and i are source vectors.zand y refer to primitive impedance and admittance matrices, which have not any information regarding connection of elements.

III.

Network Graph Formation

A typical 3-bus, 3-line network shown in figure 4 is used to define the corresponding graph.

Figure 4: A three-bus three-line power system

Buses are shown as node and each impedance is designated as branch and shown as connected line between relevant buses. For the buses with generator, a reference node is assigned. Therefore, the oriented graph of the system can be shown in figure5.

2

Formation of Ybus by Singular Transformation b2 n2

n1 b1

b4 n0

b3 n3

Figure 5: Oriented graph of the system It is worth remarking that direction of currents are arbitrary because by transpose multiplication, the desired admittance elements of final admittance matrix will be achieved.

IV.

Incidence Matrix Formation

Owning to offer the connection information of branches in network, a connection matrix with the size of (b×n), which is termed as node incidence matrix [A] will be formed to supplement primitive matrices z and y. As convention of circuit theory, the branch orientations are defined as bellow:

= +1 when the current in the branch i away from node j. = -1 when the current in the branch i enters node j. = 0 when branch i is not connected to branch j.

Therefore designated incidence matrix [A] by considering oriented graph of the system in figure5, can be formed as bellow; b 1 A= 2 3 4

n= 1

2

3

3

4

Formation of Ybus by Singular Transformation

V.

Bus Admittance Matrix Formation

Through figure 2, (I+i) = yV, for branch b,

Pre-multiplying by

, transpose of the reduced incidence matrix, which is deprived of reference

node,

However, as the algebraic sum of the currents entering each bus is zero,

This gives or, Again,

is a vector sum of all source currents entering each bus and can be designated as

with the size of (n×1).

which

is bus voltage and can be written in, [I] = [

Finally, bus admittance matrix [

][V]

] with the size of (n×n) is achieved, [

Therefore, bus admittance matrix [

] of the system depicted in figure 4 can be formed as

bellow. First the node incident matrix of the system could be formed as matrix A:

4

Formation of Ybus by Singular Transformation

A=

The matrix would be smaller by deleting the column relates to referenced bus, and then the reduced indecent matrix would be formed:

A=

Now primitive matrix impedance would be:

[Z] =

And primitive admittance matrix would be formed by inversing the impedance matrix as below:

=[ ] =

[ ] . [A] =

=

And finally admittance matrix would form:

=

=

5

Formation of Ybus by Singular Transformation

Appendix A Admittance Matrix formation of a Given Simple Power System

G1

G2

j1

j0.8

j0.4

1

2

j0.2

j0.2 3 j0.08 4

Step1: Corresponding Graph Formation of Proposed Power System b3

n1

n2 b2

b1 b5

b4 n0

n3 b6 n4

Step2: Incidence Matrix Formation

A=

6

n0

Formation of Ybus by Singular Transformation

and

Step3: Primitive Impedance and Admittance Matrices Formation

[Z] =

[Y] =

=

Step4: Admittance Matrix Formation

=

[ ][A] =

=

[

]=

=

7

Formation of Ybus by Singular Transformation

References [1] D.P.Kothari and I.J.Nagrath, Modern Power System Analysis , 3rd edition, McGraw-Hill, 2003 [2] A.Chakrabarti and S.Halder, Power System Analysis Operation and Control , 2nd edition, PHI, 2008 [3] H.Sadaat, Power System Analysis , 2nd edition, McGraw-Hill, 1999

8

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