3.7 DC Modeling

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

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

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



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.

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


Pbus,shift = (Cf − Ct)TPf,shift.

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

Pf(Θ ) = BfΘ  + Pf,shift

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