DC Modeling

3.7 DC Modeling

The DC formulation [20] is based on the same parameters, but with the following three additional simplifying assumptions.

Branches can be considered lossless. In particular, branch resistances $rs$ and charging capacitances $bc$ are negligible:
$y = ---1----≈ -1-, b ≈ 0. s rs + jxs jxs c$ (3.18)
All bus voltage magnitudes are close to 1 p.u.
$j𝜃i vi ≈ e .$ (3.19)
Voltage angle diﬀerences across branches are small enough that
$sin(𝜃f − 𝜃t − 𝜃shift) ≈ 𝜃f − 𝜃t − 𝜃shift.$ (3.20)

Substituting the ﬁrst set of assumptions regarding branch parameters from (3.18), the branch admittance matrix in (3.2) approximates to

[ ] -1-- 1τ2 − τe−j1𝜃shift Ybr ≈ jxs − -j1𝜃--- 1 . τe shift

(3.21)

Combining this and the second assumption with (3.1) yields the following approximation for $if$ :

The approximate real power ﬂow is then derived as follows, ﬁrst applying (3.19) and (3.22), then extracting the real part and applying (3.20).

As expected, given the lossless assumption, a similar derivation for the power injection at the to end of the line leads to leads to $pt = − pf$ .

The relationship between the real power ﬂows and voltage angles for an individual branch $i$ can then be summarized as

[ p ] [ 𝜃 ] f = Bibr f + Pishift pt 𝜃t

(3.24)

where

and $bi$ is deﬁned in terms of the series reactance $xis$ and tap ratio $τ i$ for branch $i$ as

b = -1--. i xisτi

For a shunt element at bus $i$ , the amount of complex power consumed is

So the vector of real power consumed by shunt elements at all buses can be approximated by

Psh ≈ Gsh.

(3.26)

With a DC model, the linear network equations relate real power to bus voltage angles, versus complex currents to complex bus voltages in the AC case. Let the $n × 1 l$ vector $Bff$ be constructed similar to $Yff$ , where the $i$ -th element is $bi$ and let $Pf,shift$ be the $nl × 1$ vector whose $i$ -th element is equal to $i − 𝜃shiftbi$ . Then the nodal real power injections can be expressed as a linear function of $Θ$ , the $nb × 1$ vector of bus voltage angles

Pbus(Θ ) = BbusΘ + Pbus,shift

(3.27)

where

Pbus,shift = (Cf − Ct)TPf,shift.

(3.28)

Similarly, the branch ﬂows at the from ends of each branch are linear functions of the bus voltage angles

Pf(Θ ) = BfΘ + Pf,shift

(3.29)

and, due to the lossless assumption, the ﬂows at the to ends are given by $Pt = − Pf$ . The construction of the system $B$ matrices is analogous to the system $Y$ matrices for the AC model:

The DC nodal power balance equations for the system can be expressed in matrix form as

gP(Θ, Pg) = BbusΘ + Pbus,shift + Pd + Gsh − CgPg = 0

(3.32)

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