3.2 Branches

All transmission lines24 , transformers and phase shifters are modeled with a common branch model, consisting of a standard π  transmission line model, with series impedance zs = rs + jxs  and total charging susceptance b
c  , in series with an ideal phase shifting transformer. The transformer, whose tap ratio has magnitude τ  and phase shift angle 𝜃shift   , is located at the from end of the branch, as shown in Figure 3-1. The parameters rs  , xs  , bc  , τ  and 𝜃shift   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 if  and it  at the from and to ends of the branch, respectively, can be expressed in terms of the 2 × 2  branch admittance matrix Ybr  and the respective terminal voltages vf  and vt

[    ]      [     ]
  if   = Y    vf   .
  it      br   vt

With the series admittance element in the π  model denoted by ys = 1∕zs  , the branch admittance matrix can be written

     [  (ys + j bc) 12 − ys-−j1𝜃--- ]
Ybr =        --21--τ       τe bshcift  .
         − ys τej𝜃shift    ys + j 2

Figure 3-1:Branch Model

If the four elements of this matrix for branch i  are labeled as follows:

      [ yi   yi  ]
Ybir =    fif   fit
        ytf  ytt

then four n ×  1
 l  vectors Y
 ff  , Y
 ft  , Y
  tf  and Y
 tt  can be constructed, where the i  -th element of each comes from the corresponding element of   i
Ybr  . Furthermore, the nl × nb  sparse connection matrices Cf  and Ct  used in building the system admittance matrices can be defined as follows. The (i,j)th   element of Cf  and the (i,k)th   element of Ct  are equal to 1 for each branch i  , where branch i  connects from bus j  to bus k  . All other elements of C
 f  and C
  t  are zero.