Search Results (3)

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented. Full article

With distributed generation (DG) being continuously connected into distribution networks, the stochastic and fluctuating nature of its power generation brings ever more problems than before, such as increasing operating costs and frequent voltage violations. However, existing robust scheduling methods of flexible resources tend to make rather conservative decisions, resulting in high operation costs. In view of this, a two-stage robust optimal scheduling method for flexible distribution networks is proposed in this paper, based on the pairwise convex hull (PWCH) uncertainty set. A two-stage robust scheduling model is first formulated considering coordination among on-load tap changers, energy storage systems and flexible distribution switches. In the first stage, the temporal correlated OLTCs and energy storage systems are globally scheduled using day-ahead forecasted DG outputs. In the second stage, FDSs are scheduled in real time in each time period based on the first-stage decisions and accurate short-term forecasted DG outputs. The spatial correlation and uncertainties of the outputs of multiple DGs are modeled based on the PWCH, such that the decision conservativeness can be reduced by cutting regions in the box with low probability of occurrence. The improved column-and-constraint generation algorithm is then used to solve the robust optimization model. Through alternating iterations of auxiliary variables and dual variables, the nonconvex bilinear terms induced by the PWCH are eliminated, and the subproblem is significantly accelerated. Test results on the 33-bus distribution system and a realistic 104-bus distribution system validate that the proposed PWCH-based method can obtain much less conservative scheduling schemes than using the box uncertainty set. Full article

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition—i.e., the parameter-space domain where it has the largest modularity relative to the input set—discarding partitions with empty domains to obtain the subset of partitions that are “admissible” candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP. Full article