Convexity of solvability set of power distribution networks

Anatoly Dymarsky, Konstantin Turitsyn

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


The solvability set of a power network-the set of all power injection vectors for which the corresponding power flow equations admit a solution-is central to power systems stability and security, as well as to the tightness of optimal power flow relaxations. Whenever the solvability set is convex, this allows for substantial simplifications of various optimization and risk assessment algorithms. In this letter, we focus on the solvability set of power distribution networks and prove convexity of the full solvability set (real and reactive powers) for tree homogeneous networks with the same r/x ratio for all elements. We also show this result can not be improved: once the network is not homogeneous, the convexity is immediately lost. It is nevertheless the case that if the network is almost homogeneous, a substantial practically important part of the solvability set is still convex. Finally, we prove convexity of real solvability set (only real powers) for any tree network as well as for purely resistive networks with arbitrary topology.

Original languageEnglish
Article number8502879
Pages (from-to)222-227
Number of pages6
JournalIEEE Control Systems Letters
Issue number1
StatePublished - Jan 2019

Bibliographical note

Publisher Copyright:
© 2017 IEEE.


  • Algebraic/geometric methods
  • Power systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Control and Optimization


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