#### 3.7 DC Modeling

The DC formulation  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)