Application of semidefinite optimization techniques to problems in electric power systems

Author / Creator

Molzahn, Daniel K., dissertant

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Online

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Summary

This dissertation investigates applications of a semidefinite programming relaxation of the power flow equations, which model the steady-state relationship between power injections and voltages in ...

This dissertation investigates applications of a semidefinite programming relaxation of the power flow equations, which model the steady-state relationship between power injections and voltages in an electric power system. In contrast to other solution techniques, semidefinite program solvers can reliably find a global optimum in polynomial time when the semidefinite relaxation is "tight." This work investigates a semidefinite relaxation of the optimal power flow (OPF) problem. This includes practical examples where the relaxation is not tight as well as computational and modeling advances required for solving large-scale, realistic OPF problems. Modeling advances include allowing multiple generators at the same bus; parallel transmission lines; and an approximate method for modeling constant impedance, constant current, constant power (ZIP) loads. Computational advances include modifying existing matrix decomposition techniques to reduce solver times, a method for obtaining an optimal voltage profile from a solution to a decomposed problem, and extension of an existing decomposition method to general networks. This dissertation also provides a sufficient condition for global optimality of a candidate OPF solution obtained by any method. This dissertation next develops a sufficient condition for infeasibility of the power flow equations. This condition is evaluated using a feasible semidefinite program. Voltage stability margins developed from this optimization problem yield measures of the distance to the power flow solvability boundary. This dissertation considers generators modeled as ideal voltage sources and generators with reactive power limits, which require either mixed-integer semidefinite programming or sum-of-squares programming formulations. This dissertation then investigates the problem of finding multiple power flow solutions. Claims in existing literature regarding the ability of a continuation-based algorithm to reliably find all power flow solutions are demonstrated to be incorrect with a counterexample system. Methods for calculating multiple power flow solutions using semidefinite programming are investigated. Although the semidefinite relaxation of the power flow equations is often "tight," non-zero relaxation gap solutions can occur. This dissertation investigates examples of non-zero relaxation gap solutions to semidefinite formulations for the optimal power flow problem, for the power flow insolvability condition, and for determining multiple solutions to the power flow equations.

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