Formation of Admittance Matrix | Power System | Electrical Engineering

In this article we will discuss about the procedure for the formation of admittance matrix in a power system.

Formation of Y_bus Using Step by Step Method:

The admittance matrix can be formed from the parameters of the system components. A diagonal element Y_ii is the sum of all admittances connected to i^th bus. An off-diagonal element Y_ik is the negative of the total admittance directly connected between i^th and k^th buses. The following step by step procedure for formation of Y_bus may be used. This procedure is very suitable.  

Let us start with [Y_bus] array set to zero. The dimensions of the Y_bus matrix is (n x n) where n is the number of buses. The total number of nodes are m = n + 1 including the ground or reference node. Consider an element having admittance y_ik connected between buses i and k. Four entries in [Y_bus] are affected – Y_ii, Y_ik, Y_ki and Y_kk.

These entries can be modified as under –  

We add the elements one by one and modify the entries of [Y_bus] according to Eq. (6.11). If an element is connected from i^th bus to reference bus, only entry Y_ii is affected.

Formation of Y_bus Using Singular Transformation:

The Y_bus can be alternatively formed by using singular transformation given by a graph theoretical approach. This alternative approach is of great theoretical and practical significance and is, therefore, discussed here.

To understand this alternative approach, the graph theory is briefly reviewed.

Graph Theory:

In order to describe the geometrical structure of the network, it is sufficient to replace the different power system components (of the corresponding power system network) such as generators, transformers and transmission lines etc. by a single line element irrespective of the characteristics of the power system components.

The geometrical interconnection of these line elements (of the corresponding power system network) is known as a graph (rather linear graph as the graph means always a linear graph). Each source and the shunt admittance across it are taken as a single element. The terminals of the elements are called the nodes.

A graph is connected if, and only if, there exists a path between every pair of nodes. A single edge or a single node is a connected graph. If every edge of the graph is assigned a direction, the graph is termed as an oriented graph. The direction is generally, so assigned as to coincide with the assumed positive direction of the current in the element.

Power networks are so structured that out of the m total nodes, one node (normally described by 0) is always at ground potential and the remaining n = (m – 1) nodes are the buses at which the source power is injected. Figure 6.5 shows the oriented graph of the network given in Fig. 6.2 (b).

A connected sub-graph containing all the nodes of the original graph but no closed path is called a tree. The tree branches form a sub-set of the elements of the connected graph. The number of branches b required to form a tree is equal to the number of buses in the network (the total number of nodes, including the reference node, is one more than the number of buses), i.e.,

b = m – 1 = n (number of buses)                                              …(6.12)

The elements of the original graph not included in the tree, form a sub-graph which may not necessarily be connected, is known as cotree. The cotree is a complement of a tree. The elements of a cotree are called the links.

The number of links I of a connected graph with e elements is given as –  

i.e., number of links equals number of elements less the number of tree branches. A tree and the corresponding cotree of graph shown in Fig. 6.5 are shown in Figs. 6.6 (a) and 6.6 (b) respectively. 

If a link is added to the tree, the corresponding graph contains one closed path called a loop. Thus a graph has as many loops as the number of links.

The above system has 9 branches. So it has 18 variables (9 branch voltages and 9 branch currents). However, it can be easily seen that all these 18 variables are not independent. The number of independent variables is found from the concept of the tree.

The number of tree branches gives the number of independent voltages. For any system the number of tree branches is equal to the number of buses. The number of links gives the number of independent current variables.

Primitive Network:

A network is constituted by many branches and each branch consists of active and/or passive elements. Fig. 6.1(a) and 6.1(b) show a network branch, containing both active and passive elements in impedance and admittance representation. The impedance is a voltage source E_rs in series with and impedance, z_rs; while in admittance form there is a current source j_rs in parallel with an admittance y_rs. The element current is I_rs and element voltage, V_rs= V_r – V_s where V_r and V_s are the voltages of the element nodes r and s, respectively.

The noteworthy point is that for steady state ac performance, all element variables (V_rs, V_r, V_s, I_rs, J_rs) are phasors and element parameters z_rs and y_rs are complex numbers.

The performance equation for impedance representation, depicted in Fig. 6.7(a), can be written as –

V_rs + E_rs = z_rs I_rs                                                            …(6.14)

and for admittance representation depicted in Fig. 6.1(b) –

I_rs + J_rs = y_rs V_rs                                                           …(6.15)

The two representations shown in Figs. 6.7(a) and 6.7(b) are equivalent wherein the parallel source current in admittance form is related to the series voltage in impedance form by –

 

 

 

A set of unconnected elements is known as primitive network. The performance equations in admittance (or impedance) form can be written for all branches.

The set of these equations in impedance form is –  

V + E = ZI                                                                       …(6.17)

and in admittance form I + J = YV                      …(6.18)

where V and E are branch voltage and source voltage matrices, I and J are branch current and source current matrices, Z is primitive impedance matrix (i.e., a matrix whose elements are branch self-impedances) and Y is primitive admittance matrix (i.e., matrix whose elements are branch self-admittances). These are related as Z = 1/Y. If there is no coupling between elements, Z and Y are diagonal matrices.

Network Variables in Bus Frame of Reference:

Linear network graph is quite helpful in the systematic assembly of a network model. The main problem in the derivation of mathematical models for large and complex power networks is to select a minimum or zero redundancy (linearly independent) set of current or voltage variables which is sufficient to provide the information about all element voltages and currents.

One set of such variables are the b tree voltages. It can easily be seen by using topological reasoning that these variables constitute a non-redundant set. The knowledge of b tree voltages allows us to compute all element voltages and therefore, all bus currents assuming all element admittances being known.

Consider a tree graph depicted in Fig. 6.6 (a) where the ground node is taken as the reference node. This is the most appropriate tree choice for a power network. With this choice the b tree branch voltages become identical with the bus voltages because the tree branches are incident to the ground mode.

Bus Incidence Matrix:

If “G” is a graph with “n” nodes and “e” elements, then the matrix A̅ whose n rows correspond to the “n” nodes (i.e., vertices) and “e” columns correspond to the “e” elements, i.e., edges, is known as an incidence matrix.

The matrix elements are:

a_ik = 1 if i^th element is incident to and directed away from the k^th node (bus).

= – 1 if the i^th element is incident to but directed towards the k^th node

= 0 if the ith element is not incident to the k^th node.

The dimension of this matrix is n x e and its rank is less than n.

Any node of the connected graph can be selected as the reference node and then the variables of the remaining n – 1 nodes which are termed as buses can be measured w.r.t. this assigned reference node.

The matrix “A” obtained from the incidence matrix A̅ by deleting the reference row (corresponding to the reference node) is termed as reduced or bus incidence matrix (the number of buses in the connected graph is equal to n – 1 where n is the number of nodes). The order of this matrix is (n –  1) x e and its rows are linearly independent with rank equal to (n – 1).

For the specific system shown in Fig. 6.5, the 9-branch voltages (V_b1, V_b2, V_b3, … V_b9) can be expressed in terms of 4-bus voltages (V₁ …. V₄) as below:

Equation (6.19) can be written in matrix form as –

V = A V_bus                                                          …(6.20)

Where the bus incidence matrix A is –  

This matrix is rectangular and, therefore, singular. Its elements a_ik are found as per rules given above.

Formulation of Y_bus and Z_bus:

Substituting Eq. (6.20) into Eq. (6.18), we have – 

I + J = Y A V_bus                                                    …(6.22) 

Premultiplying Eq. (6.22) by A^T (i.e., transpose of the bus incidence matrix) we have –

A^T I + A^T J = A^T Y A V_bus                                  …(6.23)

Each component of the n-dimensional vector A^T I is the algebraic sum of the element currents leaving the nodes 1, 2, 3, …, n.

Therefore, as per Kirchhoffs’ current law –

A^T I = 0                                                                   …(6.24)

Similarly, each component of vector A^T J can be recognized as the algebraic sum of all source currents injected into nodes 1, 2, … n. These components are therefore the bus currents.

Hence we can write –

A^T J = J_bus                                                          …(6.25)

Equation (6.23) is then simplified to –

J_bus = A^T Y A V_bus                                           …(6.26)

Comparing Eqs. (6.26) and (6.25), we have –

Y_bus = A^T Y A                                                     …(6.27)

Above Eq. (6.27) suggests the formulation of Y_bus. Since matrix A is singular, A^T YA is a singular transformation of Y. The bus incidence matrix can be obtained through a computer programme. Standard matrix multiplication and matrix transpose sub-routines can be employed to compute Y_bus using Eq. (6.27). Z_bus is the inverse of Y_bus.