Table 1.
Model locations and selected parameters [26].
Fig 1.
Apartment room model used in simulations: (A) dimensions; (B) small boxes indicate positions of 15 control points of NURBS curves.
Table 2.
Basic results from general and simplified optimisation studies.
Fig 2.
Pareto fronts for both general and simplified optimisation studies.
Yellow dots represent all shading variants simulated in the general optimisation study, while green dots represent all shading variants simulated in the simplified optimisation study. Minimum heating demand is for the base case of the model without shading, and ΔH represents the relative increase of the maximum heating demand among obtained Pareto solutions with respect to the base case. Similarly, maximum cooling demand is for the base case of the model without shading, and ΔC represents the relative decrease of the minimum cooling demand among obtained Pareto solutions with respect to the base case. Graph at the bottom of each diagram represents Euclidean distance from each Pareto solution in one Pareto front to the closest Pareto solution in the other Pareto front. As the upper bound set on figure resolution may prevent distinguishing details of these diagrams, their enlarged versions are added in S1 File for easier viewing.
Fig 3.
Shading parameters of solutions in the convex hull of Pareto fronts for simplified optimisation study.
Horizontal axis shows the ratio of efficiencies and source energy conversion factors for district cooling and heating. Vertical axis depicts depth values for the corresponding group of control points in the shading. Green horizontal steps determine parameter values: y0 = y1 for the lower part of the western fin, y2 = y3 for the upper part of the western fin, y4 for the joint of the western fin and the overhang, y5 = ⋯ = y9 for the internal part of the overhang, y10 for the joint of the overhang and the eastern fin, and y11 = ⋯ = y14 for the remaining part of the eastern fin. The steps are stretched between the values of ratios
for which the corresponding shading in the convex hull has the minimum total equivalent source energy demand. Gray vertical line is placed at r = 0.3038, which corresponds to current efficiencies and source energy conversion factors for district cooling and heating in USA [55]. Red line shows fitted arctangent function for depths of each control point group.
Table 3.
Convex hull shadings with minimal equivalent source energy for r = 0.3038, together with best shadings found in a separate genetic optimisation with equivalent source energy as its single objective.
Fig 4.
Sketchup visualisations of convex hull shadings with minimal equivalent source energy for r = 0.3038.
In the remaining six locations: Memphis, San Francisco, Chicago, Vancouver, Burlington and Duluth, minimal equivalent source energy is attained in the base case of the model without any shading.
Table 4.
Parameters of fitting functions for depths of control point groups of shadings in the convex hull of Pareto fronts for simplified optimisation study.
Fig 5.
Heating and cooling demands of the Pareto solutions for the simplified optimisation study, the corresponding convex hull shadings and simplified shadings obtained by fitting control point group depths of convex hull shadings.
Green dots represent all shading variants simulated in the simplified optimisation study. Graph at the bottom of each diagram represents Euclidean distance from each Pareto solution to the closest fitted shading. As the upper bound set on figure resolution may prevent distinguishing details of these diagrams, their enlarged versions are added in S2 File for easier viewing.