Impedance and Admittance Matrices

FORMULATION OF NODAL ADMITTANCE MATRIX

• Power system network can be analysed using either mesh current method or nodal voltage method. However, from the view point of the computation time and memory, the nodal admittance formulation employing the node voltages as the independent variable is the most economical. Hence, it is most preferred.
• In load flow study following points need to be considered :
  (i) Generators are represented by complex power sources (S_Gi = P_Gi + jQ_Gi)
  (ii) Loads are represented as complex power demand (S_Di= P_Di + jQ_Di)
  (iii) The transmission lines connected to ith bus are replaced by their π-equivalent circuits, i.e., with series admittance and half line charging admittance.
  (iv) Each transmission line will present an admittance between the bus and the ground. Hence, the total admittance (Y_io) is the sum of the admittances due to all the transmission lines connected to the bus.

Now, let us consider the figure shown below.

Let us assume that there is no mutual coupling between the transmission lines and there is no regulating transformer present in system.

Here, Y_ik=(i ≠ k) = total admittance connected between the i^th and the k^th buses [Y_ik (i ≠ k) = 0
if no transmission line is between ith and k^th bus]
Y_io=Admittance connected between i^th bus and the ground.
Now, applying KCL at the i^th bus, we have

Properties of Y_BUS:

Following are some of the properties of Y_BUS matrix.
(i) It is “sparsity matrix” “sparsity matrix” “sparsity matrix” “sparsity matrix” “sparsity matrix”.
(ii) It is a symmetric matrix i.e. [Y_BUS] = [Y_BUS]^T.
(iii) It is a square matrix (n × n) and hence it’s inverse is possible.
(iv) The inverse of a Y_BUS matrix is a full matrix, i.e., Z_BUS.

• Since in a power system each bus is connected only to a few other buses (two or three), there are large number of zero elements in a Y_BUS matrix compared to a Z_BUS matrix.
• Load flow study can also be done using a Z_BUS matrix but, it is usually done with a Y_BUS matrix because it’s sparsity is very high.