POWER SYSTEM ANALYSIS, DETERMINATION OF BUS ADMITTANCE MATRIX USING SINGULAR TRANSFORMATION METHOD

 SINGULAR TRANSFORMATIONS

The primitive network matrices are the most basic matrices and depend purely on the impedance or admittance of the individual elements. However, they do not contain any information about the behaviour of the interconnected network variables. Hence, it is necessary to transform the primitive matrices into more meaningful matrices which can relate variables of the interconnected network.

 

Bus admittance matrix, YBUS and Bus impedance matrix, ZBUS

In the bus frame of reference, the performance of the interconnected network is described by n independent nodal equations, where n is the total number of buses (n+1nodes are present, out of which one of them is designated as the reference node).

For example a 5-bus system will have 5 external buses and 1 ground/ ref. bus). The performance equation relating the bus voltages to bus current injections in bus frame of reference in admittance form is given by

IBUS = YBUS EBUS

Where EBUS = vector of bus voltages measured with respect to reference bus IBUS = Vector of currents injected into the bus

YBUS = bus admittance matrix

The performance equation of the primitive network in admittance form is given by i + j = [y] v

Pre-multiplying by At (transpose of A), we obtain

At i +At j = At [y] v

However, as per equation 

At i =0,

since it indicates a vector whose elements are the algebraic sum of element currents incident at a bus, which by Kirchhoff‟s law is zero. Similarly, At j gives the algebraic sum of all source currents incident at each bus and this is nothing but the total current injected at the bus. Hence,

At j = IBUS

Thus  we have, IBUS = At [y] v

However,  we have v =A EBUS

And hence substituting in  we get,

IBUS = At [y] A EBUS

Comparing  we obtain,

YBUS = At [y] A

The bus incidence matrix is rectangular and hence singular. Hence, (22) gives a singular transformation of the primitive admittance matrix [y]. The bus impedance matrix is given by ,

ZBUS = YBUS-1

Note: This transformation can be derived using the concept of power invariance, however, since the transformations are based purely on KCL and KVL, the transformation will obviously be power invariant.