# Structural Properties of Linearized Power Flows and Power Grid Design

• Struktureigenschaften linearisierter Lastflussprobleme und des Stromnetzplanungsproblems

Lemkens, Stephan; Koster, Arie M. C. A. (Thesis advisor); Triesch, Eberhard (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2015)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2015

Abstract

In this thesis we linearize the power flow and power grid design problem and analyze the mathematical properties of these problems. We introduce a linearization which takes care of some of the drawbacks of the well-established DC formulation. The DC formulation is an approximation of the nonlinear power flow equations, useful for deriving an approximation of the active power flow. Our new linearization, called AC-linear, is based on Taylor expansion of the nonlinear formulation and yields information on active and reactive flows. We show that it is a generalization of the often used DC formulation. Our main contribution is the thorough analysis of the newly introduced AC-linear formulation. We show the existence of combinatorial structures in the problem, which involve the class of bispanning graphs. The first part concludes with a computational comparison of the quality of the two discussed linearizations. We are able to show, that the AC-linear formulation is superior to the DC one when interested in active and reactive flow information. The second part of this work considers the impact of the new linearization on the power grid design problem. We therefore study the computational complexity of various power grid design problems and show that a special formulation of the DC problem can be considered as a partition problem without the involvement of power flow equations. We show that this does no longer hold for the generalized formulation. As the set of feasible power grid designs is given by a polytope, we thoroughly study its structure for both formulations. We then give an in depth analysis of a special case of the DC power grid design polytope, where the polytope's dimension is easily determinable. We are able to derive facets for the polytope, utilizing results of both the connected-subgraph and the knapsack polytope. We then perform a computational study in order to determine the tractability of the two linearizations on modern integer programming solvers. While the DC model significantly outperforms the AC-linear one, its computed designs do not allow for a feasible AC power flow. We show that the designs derived by using the AC-linear formulation are superior with regards to this property. Thus the newly introduced formulation allows for a more accurate approximation of the power grid design problem.

Institutions

• Discrete Optimization Teaching and Research Unit [113320]
• Department of Mathematics [110000]