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 and
charging capacitances are negligible:
(3.18) - All bus voltage magnitudes are close to 1 p.u.
(3.19) - Voltage angle diﬀerences across branches are small enough that
(3.20)

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

| (3.21) |

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

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 .

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

| (3.24) |

where

and is deﬁned in terms of the series reactance and tap ratio for branch as

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

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

| (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 vector be constructed similar to , where the -th element is and let be the vector whose -th element is equal to . Then the nodal real power injections can be expressed as a linear function of , the vector of bus voltage angles

| (3.27) |

where

| (3.28) |

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

| (3.29) |

and, due to the lossless assumption, the ﬂows at the to ends are given by . The construction of the system matrices is analogous to the system matrices for the AC model:

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

| (3.32) |