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Fractal theory application for heat demand forecasting in district heating system

Aleksiejuk P., Bujalski W.

Rynek Energii

2016

Nr 5

95--98

The aim of this thesis is to formulate short-term heat demand forecasting model based on fractal theory that predicts the value of heat demand for all receivers connected to district heating system depending on weather forecast and calendar information. The paper contains descriptions of basic fractal theory concepts, techniques, analysis of self-similarity of heat demand data, and developed methodology of constructing fractal interpolation curve. Forecast accuracy was investigated for planning heat production in 24-hour horizon.

Celem badań jest opracowanie krótkoterminowego modelu prognostycznego opartego na teorii fraktali, który prognozuje wartość zapotrzebowania na ciepło wszystkich odbiorców miejskiej sieci ciepłowniczej w zależności od progno-zy pogody oraz danych kalendarzowych. W publikacji zawarto opisy podstawowych założeń, technik, analiz samopodobieństwa danych o zapotrzebowaniu na ciepło oraz metodologii konstruowania krzywej interpolacji fraktalnej. Dokładność prognozy zweryfikowano na podstawie planowania produkcji ciepła w horyzoncie prognozy wynoszącym 24 godziny.

Interpolacja fraktalna jako stochastyczna odwrotność generalizacji kartograficznej

Olszewski R.

Polski Przegląd Kartograficzny

2001

T. 33, nr 4

339-350

W artykule pokazano możliwość zastosowania interpolacji mono- i multifraktalnej jako stochastycznej odwrotności procesu generalizacji kartograficznej. W celu lepszego zrozumienia zagadnienia omówiono podstawowe pojęcia geometrii fraktalnej.

The article attempts to apply fractal analysis in cartographic research of selected components of natural environment. Fractal description, which bases on fractal geometry and analysis, invented by B.B.Mandelbrot in the seventies, became an integrating interdisciplinary tool for many scientific domains, particularly in natural sciences. Application of procedures, which originate from this division of mathematics, can also , according to the author, add to the development of analytical cartography. A fractal is a shape, which consists of parts in a way similar to whole. Fractal geometry can therefore become a tool in describing complex shapes of natural phenomena, like clouds, shorelines, mountain ridges. The parameter, which describes geometric complexity of those shapes is called fractal dimension. Fractal dimension (Dr) of a given object characterizes its degree of complexity and the extent, to which it fills the available space. In the result of the precess of cartographic generalization some of the source information is lost. Therefore it is vital to preserve the key metric parameters of somplified objects. Fractal dimension of an object ahows, how its metric parameters change in the process of cartographic generalization. Dr parameter of generalized geometric object makes it possible to recreate their approximate shapes in the process of fractal interpolation. This method can be treated as the reversal of the process of cartographic generalization. Basing on data of limited geometric accuracy and the fractal dimension of a given phenomenon, one can model topographic details on a level of complexity corresponding to the source materials. In this case fractal interpolation can also be treated as a method of stochastic decompression of spatial data with a known Dr parameter. Shape modeling of ganeralized objects through the process of fractal interpolation introduces a certain error. Geometric object, which represent river systems or shorelines are stochastic fractals. Statistically, a self-similar object does not consist of reduced copies of its entire self, but rather of reduced copies of its parts. This means, that in the process of fractal encoding we have to allow for a certain error, caused by the lack of homogeneity of transformed objects. The encoded picture, being a set of transformation, will not be a faithful copy of the original, but rather its approximation.