Ph.D Thesis
Ph.D Student Shtof Alex Optimal Power Flow in Networks with Tree Topology Department of Industrial Engineering and Management Professor Shoham Sabach Professor Amir Beck

Abstract

We study two classes of the so-called Optimal Power Flow (OPF) problem for networks with a tree topology, also known as radial networks. For each class we devise an algorithm for obtaining an approximate globally optimal solution. To the best of our knowledge, algorithms for both problem classes which produce globally optimal solution were not previously known.

Both algorithms share a similar structure. First, we establish a relationship between the optimal solution set of the OPF problem on a given network and the optimal solution set of the OPF problem on a smaller network. Then, we use the above relationship repeatedly to connect the optimal solution set of the OPF problem on the given network with the optimal solution set of a terminal problem, whose associated OPF problem is simple enough to be quickly and reliably solved. This connection results in an algorithm for solving a given OPF problem - we construct the terminal problem, find its optimal solution, and use the established connection to obtain an optimal solution of the original problem. However, the above-mentioned relationship consists of mathematical operations which cannot be computed exactly, and thus we resort to approximation. To evaluate the effect of the approximation on the quality of the algorithm's output in terms of feasibility and sub-optimality, we perform numerical experiments on several test problems, and analyze the convergence properties of the approximation scheme.

The first family comprises networks with one free generator, while the rest of the nodes are consumers and generators whose consumption and generation parameters are known. For this family we have developed a fast, robust, and exact method. This claim is backed up by a set of numerical experiments which demonstrate the method's effectiveness. The second family comprises networks with several free generators. The resulting algorithm requires much more computational power and computer memory. However, because it can easily be implemented to run in parallel, it adapts well to the highly parallel nature of modern computers, whose parallel processors count is rapidly growing.