Bus Admittance Matrix | Power System | Electrical Engineering

A power system may comprise several buses interconnected through transmission lines. Power is injected into a bus from generators, while the loads are tapped from it. Of course, there may be buses with only generators, and there may be others with only loads.

Some buses may have both generators and loads while some others may have static capacitors (or synchronous condensers) for reactive power compensation or voltage control. The surplus power at some of the buses is transported through transmission lines to the buses deficient in power.

Single line diagram of a simple 4-bus system with generators and loads at each bus is shown in Fig. 6.1. To arrive at the network model of a power system, a short line may be represented by a series impedance, long line by a nominal π model while very long lines by equivalent π.

Line resistance is usually neglected with a small loss in accuracy but a great deal of saving in computation time. It is convenient to consider loads as negative generators and lump together the generator and load powers at the buses.

Let S_Gi denote the 3-phase complex generator power flowing into the i^th bus and S_Di denote the 3-phase complex power demand at the i^th bus.

Let S_Gi and S_Di may be represented as –

S_Gi = P_Gi + j Q_Gi

and S_Di = P_Di + j Q_Di

Net complex power injected into the bus is given as –

S_i = P_i + j Q_i

= (P_Gi – P_Di) + j (Q_Gi – Q_Di)

The real and reactive powers injected into the i^th bus are then –

P_i = P_Gi – P_Di

Q_i = Q_{Gi –} Q_Di      i = 1, 2, 3, …, n                     …(6.1)

Network model of the given power system, worked out on the above lines is shown in Fig. 6.2 (a). S₁, S₂, S₃, S₄ denote the net 3-phase complex power flowing into the buses and I_1, I₂, I₃, I₄ denote the currents flowing into the buses. Each transmission line is represented by a π circuit.

The equivalent circuit of Fig. 6.2 (a) has been redrawn in Fig. 6.2 (b) where all the sources have been shown connected to a common reference at ground potential and the shunt admittances at the buses have been lumped. Besides the ground node, it has 4 other nodes (buses) at which the current from the sources is injected into the network. The line admittance between nodes i and k is depicted by y_ik = y_ki Further, the mutual admittance between lines is assumed to be zero.

Application of kirchhoff’s current law to the four nodes of fig. 6.2(b) gives the following equation –

The above Eq. (6.2) can be rearranged and written in matrix form as below –  

The admittance terms on RHS are re-designated as – 

Matrix Eq. (6.3) is written in terms of self bus admittance Y_i and mutual bus admittance Y_ik as follows –  

From network theory we know that each admittance Y_ii is known as self-admittance (or driving point admittance of i^th node and is equal to the sum of the admittances connected to i^th node. Each off-diagonal term Y_ik is known as mutual admittance (or transfer admittance) between i^th and k^th nodes and is equal to the negative of the sum of all the admittances connected directly between i^th and k^th nodes.

Further Y_ik = Y_ki.

The Eq. (6.4) can be written in compact form as –

[I_bus] = [Y_bus] [v]                                                                                               …(6.5)

where [I] is the node current matrix, [V] is the node voltage matrix and [Y_bus] is the bus admittance matrix.

General equation for n-bus network based on Kirchhoffs’ current law and admittance form is –

[I] = [Y_bus] * [V]                                                                                              …(6.6)

where [I] is the M-bus current matrix, [V] is the w-bus voltage matrix and, [Y_bus] is called the bus admittance matrix and Eq. (6.6) is written as –

I = Y_bus V

and is called the bus admittance matrix, and V and I are the n-element node voltage matrix and node current matrix respectively.

Y_bus matrix for n-bus network has n rows and n columns.

Each of the Y terms in the rows and columns has two subscripts:

1. The first subscript refers to the bus number on which the current is expressed.

2. The second subscript refers to the bus number whose voltage has caused that current component.

3. The terms on diagonal are self-admittances.

4. All the non-diagonal terms are mutual admittances.

It is seen that the current entering bus i is given as –

 

 

Self-Admittance and Mutual Admittance Elements:

1. Self-Admittance of Node:

The terms Y_ii (i = 1, 2, 3, 4) are self-admittances of respective nodes and represent the algebraic sum of all the admittances connected to that node. Each diagonal term in the Y_bus matrix is a self-admittance term.

If all the nodes, except node i, are shorted with the slack or reference bus and current I is injected in the particular node i bus and the voltage V across that node i and the reference bus is measured then the ratio I/V gives the self-admittance of that node.

Thus, self-admittance of node 4 is (i = 4)

2. Mutual Admittance between Two Nodes:

The mutual admittance terms (or transfer admittance terms) are the terms Y_ik in Y_bus matrix. All the non-diagonal terms in the Y_bus matrix are mutual admittance terms Y_ik (i; k = 1, 2, 3, 4, … n)

Mutual admittance between two buses is the negative of the sum of all the admittances connected directly between those two buses.

Also Y_ik=Y_ki

For measuring the mutual admittance between the two nodes, all the nodes, except one of the two nodes (node i or node k), are shorted with the slack or reference bus. Current I is injected in the shorted node and the voltage V across the two nodes is measured. The ratio I/V gives the mutual admittance between the two nodes.

Thus,

Mutual admittance between nodes 1 and 2 is Y₁₂ = Y₂₁