Date Published: February 28, 2019
Publisher: Public Library of Science
Author(s): Sanja Stevanović, Dragan Stevanović, Matthias Dehmer, Long Wang.
We studied previously optimal shape of external shading of windows in a cellular office with an outer edge modeled by a non-uniform rational basis spline (NURBS) curve whose control points were placed uniformly around western fin, overhang and eastern fin of the window, and whose depths were allowed to vary independently. We observed there that for each climate considered in the study there exists a shading shape close to the optimal one, but with a substantially simpler structure of control points for the NURBS curve. This simpler structure was reflected in partitioning control points into six groups such that all control points in the same group have equal depths, with groups corresponding to lower part of the western fin, upper part of the western fin, joint of the western fin and the overhang, internal part of the overhang, joint of the overhang and the eastern fin and the remaining part of the eastern fin. Here we confirm that shadings with control point structure restricted in such way can perform as well as shadings with unrestricted control points by optimising shape of external shading of windows in an apartment room for both restricted and unrestricted control point structure for the same range of climates, and showing that differences in heating and cooling demands between Pareto optimal shadings in both cases are negligibly small. This grouping of control points thus gives a simple and natural division of shading into a small number of basic constituents that have most impact on its heating and cooling demands. We further consider the convex hull of the Pareto front for shadings with restricted control points, as it contains shadings that minimise equivalent source energy in terms of the ratio of efficiencies and source energy conversion factors for district heating and cooling. We show that, in cases when depths of control point groups in convex hull shadings do not experience sudden changes between their extremal values, these depths can be fitted reasonably well by a sigmoid function that results in functional shadings that satisfactorily approximate heating and cooling demands of shadings in the Pareto front.
The process of optimizing external shading is described in detail in subsequent subsections. To put it succintly, each control point of an outer NURBS curve has a predefined set of positions and the search space consists of NURBS curves determined by all combinations of such control point positions. Genetic algorithm, as implemented in simulation manager jEPlus+EA , is instructed to approximate Pareto front of heating and cooling demands of various shading alternatives, which are modeled through auxiliary Python package and simulated by EnergyPlus. For further analysis, convex hull of the approximated Pareto front is found by a local implementation of th Fortune’s variant of the Graham’s scan method , while fitting of control point depths in convex hull shadings is done by the Levenberg-Marquardt algorithm [29, 30], as implemented in gnuplot.
In previous study  on optimal shading of windows in a cellular office in the PNNL office building model, where the outer edge of the shading was modeled by a NURBS curve with independently varying depths of control points, we observed that in a close vicinity of each optimal shading there exist shadings with substantially simpler structure in which control points are divided in six groups corresponding to lower part of the western fin, upper part of the western fin, joint of the western fin and the overhang, internal part of the overhang, joint of the overhang and the eastern fin, and the remaining part of the eastern fin, such that all control points within the same group have equal depth. We confirm here that the same holds for shadings of windows in a room in the PNNL high-rise apartment building model by showing that the Pareto fronts of two genetic optimisation runs, one for shadings with independently varying depths of control points and another for shadings with control points divided in groups of equal depth, have very similar shapes with the root mean square Euclidean distance between two Pareto fronts being less than 1.4kWh. This allows to use simpler NURBS lined shadings, defined by a smaller number of parameters, in future studies of external window shading in both office and apartment buildings.