bv_cvxbook_extra_exercises

# E the optimal value of the problem minimize tr c t s

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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 problem m minimize 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...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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