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Optimal Power Flow
Published in J.C. Das, Power Systems Handbook: Load Flow Optimization and Optimal Power Flow, 2018
The algorithms based on Karmarkar’s projective method have polynomial-time complexity requiring O(nL) iterations. These algorithms do not appear to perform well in practice. The algorithms based on dual affine scaling methods exhibit good behavior to real-world problems. Most primal–dual barrier path following methods have been shown to require O(nL) iterations at the most. We will discuss a primal–dual IP method for LP [19,20].
Optimal Power Flow
Published in J.C. Das, Power System Analysis, 2017
The algorithms based on Karmarkar’s projective method have polynomial–time complexity requiring O(nL) iterations. These algorithms do not appear to perform well in practice. The algorithms based on dual affine scaling methods exhibit good behavior to real world problems. Most primal–dual barrier path following methods have been shown to require O(nL) iterations at the most. We will discuss a primal–dual IP method for LP [20,21].
Interior Point Methods
Published in James A. Momoh, Electric Power System Applications of Optimization, 2017
Affine-scaling methods have no known polynomial time complexity, and can require an exponential number of iterations if they are started close to the boundary of the feasible region. Also, it has been shown that these methods can make it difficult to recover dual solutions and prove optimality when there is degeneracy. However, these methods work well in practice. Very recently, a polynomial time bound for a primal-dual affine method has been obtained.
Improving a primal–dual simplex-type algorithm using interior point methods
Published in Optimization, 2018
T. Glavelis, N. Ploskas, N. Samaras
Since Karmarkar's algorithm [17], many improvements have been made both in theory and in practice of IPMs. An infeasible IPM moves from a positive point to a positive point trying to achieve feasibility and optimality, simultaneously; this is the big difference with the simplex algorithm, which follows a sequence of adjacent boundary points to an optimal solution. It has been observed that IPMs can deal much better than the simplex algorithm in large-scale sparse LPs [18]; these problems are very common in transportation and scheduling applications that have network models at their core. IPMs are also of interest from a theoretical point of view, because they have polynomial complexity. There are three main categories of IPMs: (i) affine-scaling methods, (ii) potential reduction methods, and (iii) central trajectory methods. The affine-scaling algorithm is an attractive choice due to its simplicity and its relative good performance in practice. However, its performance is sensitive to the starting point. Potential reduction methods do not have the simplicity of affine-scaling methods, but they are more attractive than affine-scaling methods. IPMs based on the central trajectory are the most useful in theory and the most used in practice.