Advances in Mathematical Physics

Volume 2015 (2015), Article ID 827238, 6 pages

http://dx.doi.org/10.1155/2015/827238

## Power Load Prediction Based on Fractal Theory

^{1}College of Electronic & Electrical Engineering, Shanghai University of Engineering Science, Shanghai 201620, China^{2}Department of Mathematics (DIPMAT), University of Salerno, 84084 Fisciano, Italy

Received 1 August 2014; Accepted 7 September 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2015 Liang Jian-Kai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The basic theories of load forecasting on the power system are summarized. Fractal theory, which is a new algorithm applied to load forecasting, is introduced. Based on the fractal dimension and fractal interpolation function theories, the correlation algorithms are applied to the model of short-term load forecasting. According to the process of load forecasting, the steps of every process are designed, including load data preprocessing, similar day selecting, short-term load forecasting, and load curve drawing. The attractor is obtained using an improved deterministic algorithm based on the fractal interpolation function, a day’s load is predicted by three days’ historical loads, the maximum relative error is within 3.7%, and the average relative error is within 1.6%. The experimental result shows the accuracy of this prediction method, which has a certain application reference value in the field of short-term load prediction.

#### 1. Introduction

Short-term load forecasting plays an important role in control and operation of the power system. People are the main consumers of electrical energy. The periodicity of production and life of the people decides the periodicity of power load. The cyclicity of power load is performed as the week and seasonal periodicity. Power load not only is cyclical but also has certain continuity, which generally does not occur in big jumps and the load curve is continuous between any two points, making it possible to predict load.

Although traditional forecasting methods, such as gray theory, expert systems theory, and fuzzy mathematics, are relatively mature, the forecast results are often unsatisfactory [1]. To further improve the prediction accuracy, we need to make some improvements on the traditional methods. In recent years, prediction experts have put forward a prediction method, which is based on particle swarm optimization extended memory and support vector regression (SVR) and a prediction method which combines support vector machines (SVM) and wavelet neural network optimization [2–4]. This improves the accuracy of prediction but has a complex computing process.

Since Mandelbrot created fractal geometry, fractals have been described in a large number of mathematical models of natural phenomena and have increasingly attracted people’s attention. Fractals as a branch of nonlinear theory have penetrated into many other branches, and fractal dimension has been widely used in image processing, data compression, fault diagnosis, voice recognition, pattern recognition, and so on [5, 6].

This study is based on the existing similar daily load forecasting method and a deep research on the fractal characteristics of the power load, which designs corresponding fractal characteristic value algorithms to achieve scientific data processing for power load.

#### 2. Fractal Theory

##### 2.1. The Fractal Dimension

Fractal theory gives a geometric definition about calculation dimension formula: a set consists of compositions which are similar and the similarity ratio can be considered as a dimension :

Since this formula can only be used in measuring the strict self-similarity geometry, the power load can adopt a similar method to approximate calculation.

Box-counting dimension is one of the most widely used dimensions. To calculate this dimension for a fractal , imagine this fractal lying on an evenly spaced grid, and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm [7].

Suppose that is the number of boxes of side length required to cover the set. Then the box-counting dimension is defined as

##### 2.2. Fractal Interpolation Function

Fractal interpolation function is a method of fractal curve construction which has the advantage that it can reflect adjacent local features. The fractal interpolation algorithm can construct an iterated function system over the entire range rather than a function, so it can maintain the most characteristic of the original sample curve and the sample interpolation points can be displayed with rich details. Because it can well reflect the characteristics of the load curve at each point, one can more accurately predict the load.

Fractal interpolation method is based on the theory of iterated function systems. If given a set of fractal interpolation function, iterated function systems (IFS) will make the attractor close to the curve of fractal interpolation function. Each in IFS is the affine transform function, which is given by the following structure formula [8, 9]:where is a coordinate, , are elements of transformation matrix, is the vertical scaling factor, and , are constant components of the transformed .

When was given, the other parameters can be expressed as

After obtaining the parameters, the IFS attractor can be obtained by deterministic iterative algorithm. With the increased number of iterations, the fitting degree of curve obtained by interpolation continues to improve and form a stable constant interpolation curve.

###### 2.2.1. The Vertical Scale Factor

When calculating the affine coefficients of IFS, we temporarily regard vertical scaling factor as a free parameter that can be empirically selected. However, parameter has an impact on interpolation result, which is closely related to the complexity of corresponding fractal interpolation function. When , IFS converges to the only attractor.

There are many methods of calculating the vertical scaling factor . In this paper, we use the analytical method to obtain parameter ; its principle is by calculating the minimum mean square error of the original and mapping function. The process is as follows [10, 11].

There is the following data sequence: ; and are two successive interpolation points; and ; then formula (3) is rewritten aswhere and the values of points correspond to the values in the original functionThe mean square error of the mapping function and the original function isSo is obtained byTo make the minimum mean square error , the partial derivative of will be zero:where

###### 2.2.2. The Method of Seeking Attractor

Attractor of IFS has a complex structure fractal diagram; the basic idea is that global and local geometric objects have self-similar structures under affine transformation. According to this principle, ultimately getting the attractor has nothing to do with the initial generator, but it depends on a set of iterative codes by affine transformation (also known as IFS code). Currently, we can use two ways to construct attractor on a computer, a deterministic algorithm (recursive algorithm) and a stochastic algorithm (random iterative algorithm) [12, 13]. Here we take an improved deterministic algorithm.

The procedure of improved deterministic algorithm is summarized in Algorithm 1.