A Multi-Level Fuzzy Linear Regression Model for Forecasting Industry Energy Demand of Iran

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Procedia - Social and Behavioral Sciences 41 ( 2012 ) 342 – 348

doi: 10.1016/j.sbspro.2012.04.039

International Conference on Leadership, Technology and Innovation Management

A Multi-Level Fuzzy Linear Regression Model for

Forecasting Industry Energy Demand of Iran

Aliyeh Kazemi

a

*, Amir Foroughi. A

b

, Mahnaz Hosseinzadeh

a

a_{Department of Industrial Management, Faculty of Management, University of Tehran, Tehran, Iran} b_{Ministry of Petroleum, Tehran, Iran}

Abstract

The aim of this paper is to develop a prediction model of energy demand of industry sector in Iran. A fuzzy-based approach is applied for the industry energy demand forecasting using socio-economic indicators. This approach is structured as a fuzzy linear regression (FLR). A multi-level FLR model is designed properly. This paper indeed proposes a multi-level FLR model by which the inputs to the ending level are obtained as outputs of the starting levels. Actual data from 1994-2008 are used to develop the multi-level FLR and illustrate capability of the approach in this regard. The estimation fuzzy problem for the model is formulated as a linear optimization problem and is solved using the linear programming based simplex method. Furthermore, having obtained the fuzzy parameters, the industry energy demand is predicted from 2011 to 2020. The results provide scientific basis for the planned development of the energy supply of industry sector in Iran.

Keywords: Energy consumption; Industry sector; Forecasting; FLR

1. Introduction

According to the economical theories and views, energy is one of the main and the most important production factors in industry sector. Predicting its future consumption is an important step in macro-planning in industry and energy sections.

Conventional regression analyzes is one of the most used statistical tools to explain the variation of a dependent variable Y in terms of the variation of explanatory variables X as: Y f( X) where f( X) is a linear function. It refers to a set of methods by which estimates are made for the model parameters from the knowledge of the values of a given input–output data set. The goal of the conventional regression analyzes is:

(a) tofind an appropriate mathematical model and

(b) to determine the best fitting coefficients of the model from the given data.

* Corresponding author: Aliyeh Kazemi E-mail address: [email protected]

© 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of The First International Conference on Leadership, Technology and Innovation ManagementOpen access under CC BY-NC-ND license.

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The use of statistical conventional regression is bounded by some strict assumptions about the given data. This model can be applied only if the given data are distributed according to a statistical model and the relation between X and Y is crisp. Overcoming such limitations, fuzzy regression is introduced which is an extension of the conventional regression and is used in estimating the relationships among variables where the available data are very limited and imprecise and variables are interacting in an uncertain, qualitative and fuzzy way [1].

During the past decades, the applications of fuzzy regression models for energy forecasting problems have been resulted in several research papers [1-3]. In 2004 a fuzzy linear regression model had been developed by Al-Kandari et al for electric load forecasting. The estimation fuzzy problem for the model is turned out to a linear optimization problem, fuzzy linear regression. It has been found using such fuzzy model; a reliable operation for the electric power system could be obtained [2]. Shakouri et al. have formulated a hybrid model by combining a fuzzy regression with a Takagi–Sugeno–Kang (TSK) model to study the effects of climate change in electricity consumption in 2008 [3]. In 2009, a flexible fuzzy regression model has been formulated by Azadeh et al. to forecast oil consumption based on standard economic indicators [1].

In this paper industry energy demand of Iran is forecasted using FLR model considering economic and social indicators for the time span 2011 to 2020. For the estimation, time series data covering the period 1994 to 2008 are used. The remaining parts of the paper are organized as follows. In section 2, FLR model is introduced. Details of the proposed forecast strategy and numerical results are described in section 3. A brief review of the paper is given in section 4.

2. Fuzzy linear regression model

Fuzzy linear regression model was proposed by Tanaka et al [4]. This method is widely applied to various applications including marketing, management and sales forecasting [5,6]. It can also be applied for energy forecasting problems.

In this section, a formulation for fuzzy linear regression estimation problem is presented. In this model, the outputs are non-fuzzy observations. Also, the inputs are non-fuzzy inputs. The base model is assumed to be a fuzzy linear function as below:

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x A A AX A X AnXn

f

y ,~ ~₀~₁ ₁~₂ ₂...~

where Ai(i 1,2,...,n) are the fuzzy coefficients in the form of

p ,i ci

where

p

i is the middle and

c

i is

the spread. The membership function for the fuzzy coefficient A~_i is shown in Fig. 1.[7]

0 i p i c i c 1 ) ( ~ a A_i a

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The membership functions for each type of

A

~

_i are assumed a triangular membership. So it can be expressed by definition 1 as:

(2) °¯ ° ® d d otherwise c p a c p c p a a A i i i i i i i , 0 , 1 ) ( ~

By applying the Extension Principle [8], the membership function of fuzzy number y~ is given by:

(3)

^

`

^

`

°¯ ° ® z otherwise a x f y a a A y y i i i i i 0 ) , ( ) ) ( ~ max(min ) ( ~ I

From (1) and (3) we get:

(4) ° ° ° ° ° ¯ °° ° ° ° ® z z

¦

0 , 0 0 0 , 0 1 0 ) ( 1 ) ( ~ 1 0 1 0 y x y x x x c c x p p y y y i i i n i i i n i i i The spread of y~ is

¦

n i i ix c 1

and the middle of y~ is

¦

n i i i x p 1 .

we seek to find the coefficients A~_i (p_i,c_i)that minimize the spread of the fuzzy output for all data sets. Eq. (5) shows the objective function [9].

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¦¦

¦

m j n i ij i n i x c c Min 1 1 1 0 ) (

and the constraints require that each observation yjhas at least h degree of belonging to ~ yy( ), that is [10] : (6) m j h y y( _j) 1,2,..., ~ _t

The degree h is specified by the user.

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(7) m j x c c h x p p y m j x c c h x p p y ij n i i ij n i i j ij n i i ij n i i j ,..., 2 , 1 , ) )( 1 ( ,..., 2 , 1 , ) )( 1 ( 1 0 1 0 1 0 1 0 d t

¦

The above analyzes leads to the following linear programming problem:

(8) 0 , 0 ,..., 2 , 1 ) )( 1 ( ,..., 2 , 1 ) )( 1 ( . . ) ( 1 0 1 0 1 0 1 0 1 1 1 0 t t d t

¦

¦¦

¦

i i ij n i i ij n i i j ij n i i ij n i i j m j n i ij i n i p c m j x c c h x p p y m j x c c h x p p y t s x c c Min

3. The multi-level FLR model development and application

In this section industry energy demand of Iran from 2011 to 2020 is forecasted regarding socio-economic indicators using a multi-level fuzzy linear regression model. The structure of the designed multi-level FLR is given in Fig. 1.

FLR4 FLR1 value added of industry sector

investment in machinery

energy demand of the industry sector FLR2

FLR3

number of industries

industry energy demand in the last year Fig.2 : Structure of the designed multi-level FLR for industry energy demand

The main FLR (FLR4) takes industry energy demand in the last year, value added of industry sector, number of industries (industries with equal or more than 50 employees) and investment in machinery as inputs and produces the industry energy demand. The inputs to ending level are obtained as outputs of the starting levels. The value added of industry sector, number of industries and investment in machinery are forecasted using FLRs. Table 1 summarizes the FLRs inputs and output.

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Table 1: FLRs inputs and output.

FLR Inputs Output

1 value added of industry sector in the last year value added of industry sector value added of industry sector in two last year

2 number of industries in the last year number of industries number of industries in two last year

3 investment in machinery in the last year investment in machinery investment in machinery in two last year

electricity price in two last year

4 value added of industry sector, number of industries industry energy demand investment in machinery and industry energy demand in the last year

By this way fuzzy linear regression models are developed for forecasting of value added of industry sector, number of industries, investment in machinery and industry energy demand. The FLR models are presented in Table 2.

Table 2: FLR models for industry energy demand forecasting. FLR FLR models

1 VAL(t) (p₀,c₀)(p₁,c₁)VAL(t1)(p₂,c₂)VAL(t2)

2 NUM(t) (p₀,c₀)(p₁,c₁)NUM(t1)(p₂,c₂)NUM(t2)

3 INV(t) (p₀,c₀)(p₁,c₁)INV(t1)(p₂,c₂)INV(t2)

4 ENG(t) (p₀,c₀)(p₁,c₁)INV(t)(p₂,c₂)VAL/NUM(t)(p₃,c₃)NUM(t)(p₄,c₄)ENG(t1)

where VAL is value added of industry sector, NUM is number of industries, INV is investment in machinery and ENG is industry energy demand

Data related with industry energy demand model are collected from Institute for International Energy Studies (IIES) and Iran Ministry of Energy. All values given for the economic variables are normalized based on the fixed prices of 1997 (1997=100).

By using the past history data (1994-2005) fuzzy linear regression equation of industry energy demand of Iran is as bellow:

(9) ) 1 ( ) 0 , 626 . 0 ( ) ( ) 0 , 017 . 0 ( ) ( / ) 8 . 0 , 0 ( ) ( ) 0 , 00008 . 0 ( )

(t INV t VAL NUM t NUM t ENGt

ENG

Average absolute error percentage (AAEP) criteria is used for validity investigation of the model. The AAEP is calculated from the following equation:

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¦

n i xi i x i x n AAEP 1 () ) ( ) ( ˆ 1

where

x

ˆ i

(

)

is the estimated data and

x

(i

)

is the actual data.

The AAEP value for test data (2004-2008) is 3.17%. That is acceptable.

For forecasting industry energy demand in future, value added of industry sector, number of industries and investment in machinery are forecasted. Then the fuzzy models are used to predict value added of industry sector, number of industries and investment in machinery from 2011 to 2020. The

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estimation of value added of industry sector, number of industries and investment in machinery are given in Fig. 3, 4 and 5. These graphs show the actual data versus the FLR results.

Fig.4 : Estimated number of industries Fig.3 : Estimated value added of industry sector

Fig.5 : Estimated investment in machinery

By using equation (9) and related data the industry energy demand is estimated. The results in the middle level in can be seen in Table 3. Industry energy demand will reach to a level of 616 MBOE in 2020.

Table 3. Estimated industry energy demand

Years industry energy demand (MBOE) Years industry energy demand (MBOE)

2011 412 2016 523 2012 434 2017 548 2013 456 2018 569 2014 478 2019 592 2015 501 2020 616 4. Conclusions

This paper focused on forecasting the annual industry energy demand of Iran regarding socio-economic indicators using multi-level fuzzy linear regression model. A multi-level fuzzy linear regression model was designed to take industry energy demand in the last year, value added of industry sector, number of industries and investment in machinery as inputs and produces industry energy demand. The value added of industry sector, number of industries and investment in machinery were forecasted using FLR models. Actual data from 1994 to 2008 were used and industry energy demand of Iran from 2011 to 2020 was forecasted.

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