6 Transformers in the Nodal Admittance Matrix

The complex nodal admittance matrix is referred to as Y_bus. In the current formulation, the admittances of Y_bus are stored in rectangular form. The real part of Y_bus (the conductance matrix) is called G_bus. The imaginary part of Y_bus (the susceptance matrix) is called B_bus. Y_bus is formed according to two simple rules.

1. The diagonal terms of Y_bus (i.e. y_ii) are the driving point admittances of the network. They are the algebraic sum of all admittances incident upon vertex i. These values include both the self admittances of incident edges (with respect to vertex i) and shunts to ground at the vertex itself.

2. The off-diagonal terms of Y_bus (i.e. y_ij where i $\neq$ j) are the transfer admittances of the edge connecting vertices i and j.

Assuming that each transformer is modeled as two vertices (representing its primary and secondary terminals) and a connecting edge in the network graph, the procedure for incorporating a transformer into the nodal admittance matrix is as follows.

1. Compute the admittance matrix of the transformer (i.e. Y_pp, Y_ss, Y_ps, and Y_sp) at the nominal tap setting. See Section 4 for details.

2. Include the transformer’s primary admittance Y_pp as its contribution to the self admittance of the primary vertex p. That is, add Y_pp to the corresponding diagonal element of the nodal admittance matrix y_pp.

3. Include the transformer’s secondary admittance Y_ss as its contribution to the self admittance of the secondary vertex s. That is, add Y_ss to the corresponding diagonal element of the nodal admittance matrix y_ss.

4. Use Y_ps as the transfer admittance of the edge from the primary vertex p to the secondary vertex s (i.e. set the corresponding off-diagonal element of the nodal admittance matrix y_ps to Y_ps).

5. Use Y_sp as the transfer admittance of the edge from the secondary vertex s to the primary vertex p (i.e. set the corresponding off-diagonal element of the nodal admittance matrix y_sp to Y_sp).

Note: Examining Equation 19 and Equation 20, you can see that asymmetries introduced into Y_bus by a transformer are due to the fact that a${}^{\star}$ is used to compute Y_ps and a is used to compute Y_sp. When a = a${}^{\star}$, Y_bus is symmetric with respect to transformers. This condition is only true when a is real. In this situation, $\delta$ is zero and there is no phase shift across the transformer.