This preview shows page 1. Sign up to view the full content.
Unformatted text preview: re given s, and want to ﬁnd p and φ that satisfy these equations.
We are interested in solutions with voltage phase angle diﬀerences that are smaller than ±90◦ .
(Under normal conditions, real power lines are never operated with voltage phase angle diﬀerences
more than ±20◦ or so.) 141 You will show that the DC power ﬂow equations can be solved by solving the convex optimization
i= j ψ j ( pj )
subject to Ap = s,
with variable p, where
u ψ j (u) = 0 sin−1 (v/κj ) dv = u sin−1 (u/κj ) + κj ( 1 − (u/κj )2 − 1), with domain dom ψj = (−κj , κj ). (The second expression will be useless in this problem.)
(a) Show that the problem above is convex.
(b) Suppose the problem above has solution p⋆ , with optimal dual variable ν ⋆ associated with the
equality constraint Ap = s. Show that p⋆ , φ = ν ⋆ solves the DC power ﬂow equation. Hint.
Write out the optimality conditions for the problem above.
16.7 Power transmission with losses. A power transmission grid is modele...
View Full Document
- Fall '13
- The Aeneid