FORECASTING OF ELECTRICITY DEMAND IN RURAL AREAS. PART II. COMPARISON OF APPLICABILITY OF ARIMA AND TAKAGI-SUGENO 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 compares applicability of stochastic models of ARIMA type and Takagi-Sugeno fuzzy models for forecasting the monthly demand for electric energy among recipients in rural areas. It was concluded that although accuracy of forecasts set on the basis of ARIMA model is greater than in the case of Takagi-Sugeno models, fuzzy models are better in reflecting the demand for electric energy, and achievement of good quality forecasts requires much less data from the history of the process forecasted than in the case of ARIMA models.

Key words: electric energy, forecast, ARIMA models, Takagi-Sugeno models.

INTRODUCTION

Process of forecasting is closely related to uncertainty. It is most frequently described using the theory of probability. Then randomness is the only form of describing all uncertainty. The issue is treated differently by the theory of fuzzy sets, where fuzziness is a type of deterministic uncertainty. These two different forms of describing uncertainty generate qualitatively different forecasting models, and for that reason it seems worth comparing their applicability to forecasting demands for electric energy.

Forecasts of demand for electric energy are prepared by energy companies dealing with energy transfer and distribution. In particular, they prepare forecasts of monthly energy demand, and on their basis place orders on the wholesale market for energy procurement.

Therefore, the aim of this work was to compare applicability of selected stochastic and fuzzy models for forecasting of monthly demand for electric energy, whereas the analysis was limited to checking applicability of ARIMA stochastic model and Takagi-Sugeno fuzzy model, as these models seem to be particularly applicable for prediction of this type of processes [5, 6, 8, 12, 13, 14]. The aim of the paper was achieved on the basis of data concerning monthly energy consumption by recipients in rural areas, as a characteristic group of energy users.

MATERIAL AND METHODS

Calculations necessary for preparation of forecasting models and comparison of their applicability were performed on the basis of reports on monthly sales of electric energy in the years 1993-2002 to recipients in rural areas of the Małopolska Province, presented in the paper [13]. The data refers to 291,000 recipients. This number includes about 125,000 farms, of which 40% does not exceed the area of 2 ha. Current average monthly consumption of electric energy by a statistical recipient from a rural area in the Małopolska Province remains at the level of 200 kWh, varying from ca. 180 kWh in September to 240 kWh in December. In the case of farms, the variables adopt values of respectively about 190, 185 and 200 kWh/month.

Out of all data concerning monthly electricity consumption, the last 12 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. Just in the paper [13] the accuracy and the reliability of forecasts were measured ex post using the mean absolute percentage errors (MAPE_1 and MAPE_2).

ARIMA models are mixed models for autoregression and moving average, and according to the notation introduced by their authors Box and Jenkins [1], they are defined as ARIMA(p,d,q) models, or ARIMA(p,d,q)(P,D,Q)_S when stochastic processes feature periodic variances, where: p, P – autoregressive parameters, d, D – differentiation orders, q, Q – moving average parameters, S – periodicity. In the analysis, for construction of the model, software STATISTICA PL for Windows [10] was used.

ARIMA methodology became very popular in many areas, including forecasting energy demand [5, 6]. Unlike ARIMA models, fuzzy models are not used in practice for prediction of demand for electric energy, although results of studies are known concerning successful application of the theory of fuzzy sets to make forecasts in the electric energy sector [4].

For forecasts purposes, the best suited models are the ones with Takagi-Sugeno logic [8, 14]. These are predictive models allowing for setting the envisaged value of the output provided that the values of present and past inputs and outputs are known.

The fuzzy logic method proposed by Sugeno et al. [11] is based on the base of rules of a special format, which is marked with function:

IF (x₁ = A₁₁) AND ... AND (x_r = A_1r) THEN (y = f₁ (x₁, ..., x_r))          (1)
ALSO
...
ALSO
IF (x₁ = A_m1) AND ... AND (x_r = A_mr) THEN (y = f_n (x₁, ..., x_r))

where:
x₁, ..., x_r – data input;
A₁₁, ..., A_mr – fuzzy subsets of respective sets X₁, ..., X_r;
y – data output;
f₁ (x₁, ..., x_r), ..., f_n (x₁, ..., x_r) – linear functions.

In order to define the optimum model, its parameters were set using the conjugate gradient method in such a way that the total square error of the model be kept to the minimum [3, 7].

RESULTS

ARIMA models. Because ARIMA models can only be used for modelling stationary time series, and the analyzed electric energy consumption is a non-stationary process, characterized with a development tendency and multiplicative seasonability, in order to achieve stationary character of the process, it was necessary to apply seasonal and non-seasonal differentiation of the original series (d = 1, D = 1). The number and type of model parameters were identified on the basis of a diagram of empirical series and differentiated series, correlograms of process autocorrelation and partial autocorrelation. Model parameters were estimated using the highest credibility method, and the final adoption of the model was decided upon by importance of parameters, randomness of remainders and normality of model remainders distribution.

As a result of calculations and analyses performed, it was concluded that for description of monthly demand for electric energy by recipients in rural areas the model of moving average of the first order is well suited (q = 1), with periodical component of first order (Q = 1), with periodicity S = 12, i.e. ARIMA (0, 1, 1) (0, 1, 1)₁₂, which has general form:

(1 – B)(1 – B^S) z_t = (1 – θB)(1 – ΘB^S) a_t          (2)

where:
B – shift operator backwards;
1 – B – differential operator backwards;
z_t – stochastic component;
1 – θB, 1 – ΘB^S – moving average reversible operators;
θ, Θ – parameters of the moving average model;
a_t – random variable.

Results of estimation of model parameters are presented in table 1. Model preparation involved the use of 108 data (data for the period of 9 years). This is the smallest quantity of observations necessary for estimation of a model of this type.

Takagi-Sugeno models. Due to the character of variability of monthly demand for electric energy, the analysis involved construction of fuzzy models describing the function x_t = f(x_t-1, x_t-12), where x_t stands for electric energy consumption in month t. In the fuzzy rules (1) the following assumption were made: r = 2, m = 2, n = 4 and y = x_t, x₁ = x_t-1, x₂ = x_t-12. Therefore the analysis concerned models of systems with two fuzzy inputs X₁ and X₂, and one fuzzy output Y [7]. Geometrically, rules of the Takagi-Sugeno model correspond to the approximation of representation X₁ × X₂ ® Y using the sectional linear function. Each of the sets X₁ and X₂, which are closed intervals, has been divided into two fuzzy subsets with trapezoid membership functions, meeting the condition of unit division. Previous studies performed by the authors have proved that adoption of a greater number of subsets does not improve the quality of forecasts, yet significantly hinders optimization.

Six models of this type were constructed (TS9, TS8, ..., TS4), using data on monthly consumption of electric energy from 9, 8, 7, 6, 5 and 4 years respectively. In these models, the universe of discourse of inputs were different, starting with X₁ = [41.61,82.22] and X₂ = [41.61, 82.22] for model TS9, and ending with X₁ = [53.17, 82.22] and X₂ = [49.06, 82.22] for model TS4. Membership functions and general form of the base of rules for the models analyzed are presented in figure 1.

On the basis of the ARIMA and Takagi-Sugeno models developed, the monthly electricity consumption was forecast. After every forecast is calculated, its reliability and accuracy must be checked [2]. In this study, the reliability of forecasts was assessed by observing the curve of ex post forecasts of electricity demand in the months for which prediction models were built, and on the basis of their absolute percentage errors MAPE_2 [13].

As the prediction error generally grows along with the increasing length of the forecast period, data from 1999-2001 (36 values) was used to compare the reliability of forecasts. The differences between the real curve of the electricity consumption and the forecast courses are shown in figure 2, while the empirical distribution functions of relative error modules of particular forecasts are shown in figure 3. These figures indicate that forecasts made using fuzzy models are much more reliable than those calculated using the ARIMA model. This is confirmed by the mean annual values of absolute percentage error of ex post forecasts MAPE_2 which are shown in table 2. For comparison, this table also presents the absolute error values for ex post forecasts calculated as the mean values for particular months of the year. The season of the year was found to have an observable impact on the accuracy of forecasts. Forecast errors were usually greater in the winter, which results from significant differences between the winter electricity consumption of different years.

On the basis of models constructed, forecasts of electric energy consumption were made for particular months of 2002 (fig. 4), and values of mean absolute percentage errors MAPE_1 [13] were calculated for ex post forecasts on the basis of particular fuzzy models, as presented in table 3. As results from table 3, use of data from the past 5 years for construction of such models yields sufficiently accurate forecasts. Results of estimations of TS5 model are presented in table 4.

CONCLUSIONS

Forecasts of monthly demand for electric energy by recipients in rural areas, made on the basis of ARIMA model, are characterized with greater accuracy than the ones made on the basis of Takagi-Sugeno models. Mean absolute percentage error MAPE_1 of ex post forecasts set on the basis of ARIMA model amounts to ca. 2.3% and is at least by 1.5% smaller than the one set on the basis of fuzzy models. However, Takagi-Sugeno models better reflect the analyzed series of monthly consumption of electric energy, which is confirmed by smaller mean values of errors MAPE_2, which errors are recommended for assessment of forecast reliability [9]. For example, for Takagi-Sugeno TS5 model the error amounts to 3.7%, whereas for ARIMA model it is 7.6%. When one also considers the need to have the data from at least 9 years for the purposes of construction of an ARIMA model, whereas for construction of a fuzzy model yielding good quality forecasts data from only 5 years are needed, it can be concluded that applicability of Takagi-Sugeno models for forecasting monthly consumption of electric energy by recipients in rural areas seems competitive to ARIMA models and worth recommending to apply for the purposes of forecasting.

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