Branches

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

(3.1)

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

(3.2)

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

(3.3)

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.

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