FORECASTING OF ELECTRICITY DEMAND IN RURAL AREAS. PART I. APPLICATION OF FUZZY MODELS

Małgorzata Trojanowska¹, Jerzy Małopolski^2
1 Department of Power Engineering and Agricultural Processes Automation, Agricultural University of Cracow, Poland
² Department of Engineering and IT in Agriculture, Agricultural University of Cracow, Poland

ABSTRACT

The article analyzes applicability of Takagi-Sugeno fuzzy models for forecasting monthly demand for electric energy among recipients in rural areas. In particular, quality of forecasts was evaluated, made on the basis of two prepared linear fuzzy models differing with the number of fuzzy rules. It was concluded that forecasts calculated on the basis of both these models can be considered permissible, and the models can be used for prediction of monthly demand for electricity, whereas greater accuracy was achieved by forecasts made on the basis of the model described with four fuzzy rules.

Key words: electric energy, forecast, fuzzy sets.

INTRODUCTION

Distribution companies dealing with transmission and distribution of electric energy conclude contracts on purchase of electric energy on the energy market. In particular, they submit offers concerning monthly demand. Ordering incorrect amount of energy increases the costs of customer service, and in the long term has a negative impact on the quality of services delivered by the company. This may have especially unfavourable impact on the quality of services delivered to recipients in the rural areas, as even without this actual unit costs of supply of electricity in rural areas exceed its selling price [3].

For the purposes of contracts concluded, companies prepare forecasts of electric energy demand. Restructuring of Polish economy accompanied with restructuring of the electric energy sector have caused significant deterioration of credibility of forecasts made on the basis of previously applied forecasting techniques [6]. Strive for improving the quality of such forecasts has resulted in interest in alternative forecasting methods, such as artificial intelligence methods.

Among the artificial intelligence methods, forecasts of electric energy demand most frequently apply artificial neural networks [4, 5, 9, 10]. Not much attention has been given, however, to application of fuzzy sets theory for prognostics, although it is a good mathematical tool allowing the description of uncertainty and inaccuracy of input data.

The aim of this paper was to apply the fuzzy sets theory to forecast electricity demand in rural areas, and in particular to prepare fuzzy models for monthly sales of electricity and analysis of their applicability for prediction.

MATERIAL AND METHODS

A data set of monthly sales of electricity in the years 1993-2002 was acquired from a selected distribution company currently supplying energy to ca. 291,000 recipients in rural areas of the Małopolska Province. Recipients in rural areas serviced by the selected company mainly include households and farms (263,000). Average monthly electric energy consumption by a rural household in 2002 was at the level of 2,3 MWh and was by 15% higher than in 1993.

The course of monthly electric energy consumption in the years 1993-2002 by the analyzed recipients in rural areas is shown in figure 1. Out of 120 data acquired from the company, 12 last data were used for verifying accuracy of forecasts, while the data on earlier electric energy consumption were used for building the forecasting models and to check the reliability of forecasts.

The accuracy of forecasts was measured ex post using the mean absolute percentage error (MAPE) of forecasts calculated for periods n+1, …, T [1]:

(1)

where y_t – the actual value of the forecast variable for the t period;
y_t^* – the expected value of the forecast variable for the t period;
n – the number of observations used to make the forecast;
n+1,..., T – the range of empirical forecast validation.

The reliability of forecasts was assessed by means of an ex post identification of forecasts calculated for periods preceding the period of the forecast being evaluated, using the following relationship [1]:

(2)

The most frequent fuzzy models include the Mamdani models and Takagi-Sugeno models. The Mamdani models are based on the set of IF-THEN rules and constitute a qualitative description of the system that is closest to the natural language. Hence their broad application, particularly in regulation systems. A known drawback of these models is that they cannot comprise for objective knowledge on the system in an open form, thus they are not suitable for forecasting electricity demand. This drawback is not shared by models proposed by Sugeno et al. [11]. The models, known as Takagi-Sugeno models or Takagi-Sugeno-Kanga models, are linear fuzzy models that are most popular on modelling systems on the basis of experimental discrete data.

For the purposes of forecasting monthly electricity sales, the paper analyzed linear fuzzy models of systems with two inputs and one output, forming part of the so-called MISO systems [8]. The universe of discourse, which constitute sets of all numerical values that can be adopted by variables in the system discussed, have been marked for particular inputs through X₁ and X₂, where sets X₁, X₂ are closed intervals. Each of these sets was divided into fuzzy subsets (set X₁ into subsets A₁₁, ..., A_1k, set X₂ into subsets A₂₁, ..., A_2n) with trapezium-shaped membership functions meeting the condition of unit division:

(3)

where:
x₁, x₂ – input data;
– membership function of fuzzy set A_1i;
– membership function of fuzzy set A_2j.

The process of formulating a linear fuzzy model of the system includes fuzzification, inference and defuzzification. Fuzzification is an operation of calculating the degree of membership of input variables to particular fuzzy subsets, according to the following formula:

i = 1, ..., k ; j = 1, ..., n

(4)

Inference, or fuzzy logic, consists in setting the value of output data on the basis of fuzzy input data, using the set of fuzzy rules. In the case analyzed, fuzzy rules have the following form:

IF (x₁ = A_1i) AND (x₂ = A_2j) THEN (y = a₀ + a₁x₁ + a₂x₂)   (5)

where:
x_1, x₂ – input data;
y – output data;
A_1i, A_2j – fuzzy sets defined in the universe of discourse, respectively X₁ and X₂ of particular inputs;
a₀, a₁, a₂ – linear function coefficients.

Degree of conclusion activation of such a rule is equal to . If the base of rules is composed of m rules, and x₁, x₂ are system inputs, then system output y (defuzzification) is calculated as weighted average of the value of functions occurring in conclusions of rules at the point (x₁, x₂), where weights are formed of degrees of conclusion activation of the relevant rules.

In order to define the optimum model, its parameters must be set in such a way so that the total square error of the model be kept to the minimum. For this purpose, the analysis applied the conjugate gradient method [2, 7].

RESULTS

When defining the model, an important problem is setting the appropriate number of fuzzy sets and rules [8]. Their increase usually allows for achievement of greater accuracy of the model. However, its too complex structure may considerably hinder optimization. Such a model may yield worse forecasts due to unnecessary modelling of measurement noise.

In this study, Takagi-Sugeno models were built by splitting each universe of discourse into two fuzzy sets described by four fuzzy rules (MODEL 1) or into three fuzzy sets described by seven fuzzy rules (MODEL 2). In the fuzzy rules, due to the nature of the variability of the monthly electricity sales, the following assumptions were made: y = x_t, x₁ = x_t-1, x₂ = x_t-12 (where x_t stands for electricity consumption in month t), using data from 9, 8 and 7 years (i.e. 108, 96 or 84 monthly values of rural consumers’ demand for electricity) for calculation. If less data were used to construct the models, MODEL 2 could not be properly built.

Calculated results were presented using the example of models developed on the basis of data on monthly electricity sales in 1995-2001. In these models, universe of discourse X₁ is closed interval [42.49, 82.22], universe of discourse X₂ is the set [41.61, 82.22], and membership functions and base of rules for particular models have been presented in figures 2 and 3.

The models developed were then used to make ex post forecasts of electricity consumptions in the months for which predictive models had been constructed and the mean absolute percentage errors MAPE_2 of these forecasts were calculated. In the first case, the error amounted to 5.67%, while in the second case – 5.49%. On the basis of models prepared, forecasts for electricity consumption were also made for 2002 (fig. 4). Mean absolute percentage errors MAPE_1 for ex post forecasts amounted respectively to 3.76% and 4.65%, and both forecasts were slightly in excess of actual values. Similar differences between the values of MAPE_1 and MAPE_2 errors were obtained for ex post forecasts made using models built on the basis of data for 1993-2001 and 1994-2001.

CONCLUSIONS

Mean absolute percentage errors for ex post forecasts MAPE_2, set on the basis of created Takagi-Sugeno models, remain at the level of 5.5-6.0%. Therefore, the prepared forecasts can be considered permissible, and the models built can be applied for prediction of monthly demand for electric energy by the analyzed recipients in rural areas.

The more accurate forecast was the one prepared on the basis of MODEL 1. The MAPE_1 errors of forecasts made for the year 2002 using this model were significantly smaller than the mean errors of forecasts run using MODEL 2 and did not exceed 4%, which is a satisfactory value for the forecasting officer. When one also considers the more difficult optimization of MODEL 2, related to its more complex structure, it must be stated that MODEL 1 is better suited to forecast monthly demand for electric energy by recipients in rural areas.

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