All transmission lines24 , transformers and phase shifters are modeled with a common branch model, consisting of a standard transmission line model, with series impedance and total charging susceptance , in series with an ideal phase shifting transformer. The transformer, whose tap ratio has magnitude and phase shift angle , is located at the from end of the branch, as shown in Figure 3-1. The parameters , , , and are speciﬁed directly in columns BR_R (3), BR_X (4), BR_B (5), TAP (9) and SHIFT (10), respectively, of the corresponding row of the branch matrix.25
The complex current injections and at the from and to ends of the branch, respectively, can be expressed in terms of the branch admittance matrix and the respective terminal voltages and
With the series admittance element in the model denoted by , the branch admittance matrix can be written
If the four elements of this matrix for branch are labeled as follows:
then four vectors , , and can be constructed, where the -th element of each comes from the corresponding element of . Furthermore, the sparse connection matrices and used in building the system admittance matrices can be deﬁned as follows. The element of and the element of are equal to 1 for each branch , where branch connects from bus to bus . All other elements of and are zero.