# 6 Transformers in the Nodal Admittance Matrix

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

1. The diagonal terms of Ybus (i.e. yii) 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 Ybus (i.e. yij 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. Ypp, Yss, Yps, and Ysp) at the nominal tap setting. See Section 4 for details.

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

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

4. Use Yps 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 yps to Yps).

5. Use Ysp 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 ysp to Ysp).

Note: Examining Equation 19 and Equation 20, you can see that asymmetries introduced into Ybus by a transformer are due to the fact that a${}^{\star}$ is used to compute Yps and a is used to compute Ysp. When a = a${}^{\star}$, Ybus 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.