A differential analysis of the power flow equations

Krishnamurthy Dvijotham, Michael Chertkov, Steven Low

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Scopus citations


The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.

Original languageEnglish (US)
Title of host publication54rd IEEE Conference on Decision and Control,CDC 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages8
ISBN (Electronic)9781479978861
StatePublished - Feb 8 2015
Externally publishedYes
Event54th IEEE Conference on Decision and Control, CDC 2015 - Osaka, Japan
Duration: Dec 15 2015Dec 18 2015

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume54rd IEEE Conference on Decision and Control,CDC 2015
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Conference54th IEEE Conference on Decision and Control, CDC 2015

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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