- Title
- Mixed-integer quadratically-constrained programming, piecewise-linear approximation and error analysis with applications in power flow
- Creator
- Foster, James Daniel
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2014
- Description
- Research Doctorate - Doctor of Philosophy (PhD)
- Description
- This thesis investigates of the structure and solution of quadratically constrained optimization problems incorporating models of electrical power flow through transmission networks. Approximation of the equations and inequalities describing electrical power flow play a central role. Mixed-integer nonlinear programming (MINLP) theory is applied to determining good bounds to the optimal values of power flow optimization problems. After reviewing the literature on power system planning and design methods using mixed-integer programming, we apply recently developed MINLP software tools to the problem of distributed generation design in a distribution network. We investigate the merits of three mixed-integer programming approaches for finding good designs for new generator installation, along with a method for determining lower bounds on the optimal design objective by solving a knapsack problem. We then turn our attention to a typical continuous optimization problem with power flow constraints with the objective of minimising power loss. The natural question is how to construct good approximations of the equations and inequalities describing power flow. The analytic derivation of the error in the solutions of an approximate system of equations is considered. We then give a united analysis of the special structure of certain nonconvex quadratic functions that appear within the power flow constraints. It is seen that these quadratic functions can be reduced to the symmetric paraboloid function over the real plane by a linear eigenvector-based transformation of variables. Techniques are then presented for exploiting this special paraboloid structure. A systematic study of piecewise-linear approximations to this fundamental paraboloid function is set forth through the framework of the Delaunay triangulation, an object possessing a rich theory from the area of computational geometry. We consider piecewise-linear functions which interpolate the paraboloid function at the vertices of partitions of convex regions, and propose a theory for the optimization of partition geometry. Such partition optimization is over families of partitions possessing the same topology and a fixed number of polytopes, with the objective that an optimal partition minimises the maximum separation between the paraboloid function and the piecewise-linear function generated from the partition. We finally present a novel global optimization approach to forming strong outer approximating convex programs of nonconvex power flow optimization problems. This is achieved through replacing reverse convex quadratic inequality constraints in the transformed problem with a linear mixed-integer model based on piecewise-linear functions interpolating the paraboloid function. Our approach enables the computation of lower bounds to the value of the global optimum of power flow optimization problems with convex objectives.
- Subject
- optimization; mixed-integer nonlinear programming; quadratic programming; electrical power systems; optimal power flow; distributed generation; quadratic function
- Identifier
- http://hdl.handle.net/1959.13/1048192
- Identifier
- uon:14886
- Rights
- Copyright 2014 James Daniel Foster
- Language
- eng
- Full Text
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| View Details Download | ATTACHMENT02 | Thesis | 3 MB | Adobe Acrobat PDF | View Details Download |