Forecasting Electricity Consumption Using the Second-Order Fuzzy Time Series

There is a great development of Universiti Tun Hussein Onn Malaysia (UTHM) infrastructure since its formation in 1993. The development will be accompanied by the increasing demand for electricity. Hence, there is a need to forecast UTHM electricity consumption accurately so that UTHM can plan for future energy demand and utility saving decisions. Previous studies on UTHM electricity consumption prediction have been carried out using time series models, multiple linear regression and first-order fuzzy time series (FTS). The first-order FTS yield the best accuracy among these three methods. Previous forecasting problem showed higher order FTS can yield better accuracy. Therefore, in this study, the second-order FTS with trapezoidal membership function was implemented on the UTHM monthly electricity consumption from January 2009 to December 2018 to forecast January to December 2019 monthly electricity consumption. The procedure of the FTS and trapezoidal membership function was described together with January data. The second-order FTS forecast UTHM electricity consumption better than the first-order FTS.


Introduction
Time series is a sequence of equally spaced discrete temporal data. It may consist of some or all of the components such as trend, cyclical, seasonal and irregular. A trend is a long term pattern, while a cyclical is a repeated up and down movements in a time series. On the other hand, seasonal is a regular fluctuation during the same month or quarter whereas irregular is unexplained random component [1].
Forecasting is predicting future values based on the trends of past and current time series data. Forecasting for future electricity consumption is crucial for future power system planning and control. Forecasting can be divided into short-term forecasting (STF), medium-term forecasting (MTF) and long-term forecasting (LTF). STF up to one day or several weeks for scheduling the generation and transmission of electricity, MTF ranges from one day to several months to plan the fuel purchases, whereas LTF forecasts more than a year ahead up to twenty 20 years for power system planning [2][3]. The concept of fuzzy set theory was first proposed by Zadeh [4] in 1965. Based on Zadeh's [5][6][7][8] works, Song and Chissom [9] is the first to apply concept of the fuzzy set on time series and develop a first-order time-invariant Fuzzy time series (FTS) model in 1993. Their definition of FTS is as follows: Let Y (t) (t = . . . , 0, 1, 2, . . .), be a time series , a subset of R and be the universe of discourse on which fuzzy sets f i (t) (i = 1, 2, . . .) are defined. Let F(t) be a collection of f i (t). Then, F(t) is called a fuzzy time series on Y (t) (t = . . . , 0, 1, 2, . . .). Song and Chissom later applied time-invariant FTS [10] and time-variant [11] FTS on the enrolment of Alabama University from the years 1971-1992. Their proposed procedure using FTS to forecast are as follow: 1. Define the universe of discourse U = [D rnin -D 1 , D max + D 2 ]. 2. Partition the universe of discourse into several even equal length intervals as u l , u 2 , ..., u rn . 3. Define some fuzzy sets on the universe. where × is the min operator, T is the transpose operator and ∪ is the union operator. 6. Forecasted output using where A i -1 and A i are the fuzzified enrollments of year 1 i  and i represented by a fuzzy set. The symbol denotes the Max-Min composition operator. 7. Defuzzify forecasted results.
Chen [12] simplified Song and Chissom [10][11] procedure to implement first-order FTS with triangular Fuzzy membership function on the enrolment of Alabama University from the years 1971-1992. Chen [12] work has sparked the researchers' interests to improve the accuracy of FTS forecasting. Eleruja, Mu'azu and Dajab [13] improved the determination of the universe of discourse by replacing D 1 and D 2 [10][11] using the revised standard deviation while Cheng, Chang and Yeh [14] using standard deviation. The length of intervals was studied by Huarng [15]. Poulsen [16] and Tay [17] used trapezoidal membership functions of fuzzy time series instead of the triangular membership function. Later, researchers [16,[18][19] proved that higher order FTS improve the accuracy of the forecasting.
Konica and Hanelli [20] adopted time, historical and forecasting value of the temperature and previous day load as the input of fuzzy interference system toolbox in Matlab for a short-term load forecasting electricity consumption for Albania. They predicted the next-day electricity consumption.
University Tun Hussein Onn Malaysia (UTHM) is a developing Malaysian Technical University in south peninsular Malaysia. UTHM is located in Batu Pahat, Johor Malaysia. It has great development since its formation in 1993. There is a new campus set up in Pagoh, Johor in 2017. The development is accompanied by the usage of electricity consumption.
Forecasting of UTHM electricity consumption has been studied using time series analysis [21], multiple linear regression [22] and first-order FTS [17]. The forecasting work [17,21] utilized monthly data from January 2011 to December 2017, while [22] used monthly data from January 2011 to August 2018. The mean absolute percentage errors (MAPEs) are 5.74%, 11.14% and 10.62% for the study of [17,[21][22] respectively. Accurate electricity consumption forecasting is important for Development and Maintenance Office, UTHM. Based on the future forecast, they can plan for the strategy to reduce the electricity consumption to save on the electricity bill due to financial constraints facing by UTHM. Besides, they can apply for appropriate next year budget on electricity from bursary of UTHM based on the accurate future forecast. Hence, there is a need to reduce the MAPE obtained by previous works of UTHM electricity forecasting. From the previous three studies, it is noticed that FTS yields the smallest error of 5.74% and previous studies [16,[18][19] showed higher order FTS can yield better accuracy. Therefore, in this study, the second-order FTS with trapezoidal membership function was applied on monthly UTHM electricity consumption from January 2009 to December 2018 and year 2019 electricity consumption was forecasted. The accuracy of the second-order FTS was later compared with the first-order FTS.

Fuzzy time series (FTS)
The input of this study is the same month of electricity consumption from the year 2009-2018, for example with the input of January electricity consumption from 2009-2018, the January 2019 electricity consumption will be forecasted. The process was repeated for February, March, … till December. The following steps show the general procedure to be taken in order to forecast monthly 2019 electricity consumption.
Sort the values of the same month electricity consumption from 2009-2018 in ascending order. 1. Compute distance, D i between any two consecutive electricity consumptions in the sorted dataset as Find average distance, AD between any two consecutive electricity consumptions in the sorted dataset as 8. When data belong to two fuzzy sets, the trapezoidal Fuzzy number (6) will be used to find the membership degree. The highest membership degree will be used to determine the membership of data.
Next, a Fuzzy set relationship will be determined. If the time series variable F(t-2), F(t-1) and F(t) are fuzzified as A i , A j , and A k respectively, then the first order Fuzzy set relationship is A i 10. The fuzzy linear relation group (FLRG) will be determined by grouping the same fuzzy set which is related to more than one set. 11. The midpoint of each of the FLRG will be computed. 12. The forecasted value will be the average value of the midpoint of the FLRG values.

Error analysis
The performance of the above time series methods can be measured by mean absolute percentage error (MAPE) as below: where ,y ii y are real and forecasted data respectively, n is the number of real data.

Results and Discussion
The forecasting process of January 2019 electricity consumption will be started on the January data from the year 2009 to 2018 which is shown in table 1 as the following steps: Second-order FTS: Next, a fuzzy logical relation group (FLRG) were determined by grouping the same group of fuzzy set relationships as shown in table 4 for first-order FTS and table 5 for second order FTS.       Table 9 gives the MAPE values for first-order and second-order FTS forecasting from January 2009-December 2018. The MAPE of the second-order FTS is 3.627 % which is much lower than the firstorder FTS which is 7.240 %. Hence, in this case, the second-order FTS has a higher accuracy of prediction as compared to the first-order FTS.  1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5   The secondorder FTS 3.627 Table 10 gives the first four-month 2019 UTHM real electricity consumption recently collected and 2019 predicted values by the first and second-order FTS. The MAPE values for the first and second-order FTS between the first four months of 2019 data is shown in Table 11. It is shown that for future forecast, the second-order FTS still performs better as it gives a smaller error of MAPE of 4.376 if compared to 5.124 for the first-order FTS. The overall MAPE from January 2009 to Apr 2019 for both the first and second-order FTS is given in Table 12. Again it is proved that the second-order FTS still performs better as it gives a smaller error of MAPE of 3.657 if compared to 7.164 for the first-order FTS.

Conclusion
The first-order and second-order FTS using trapezoidal membership function and revised average distance to replace arbitrary numbers of D 1 nd D 2 [10][11] was applied on monthly UTHM electricity consumption from January 2009-December 2018 to forecast monthly 2019 UTHM electricity consumption. The same month data was used to forecast the next year electricity consumption. The January data was utilised and a step-by-step procedure for the first-order and second-order FTS was demonstrated. The procedure was repeated twelve times for twelve months. Based on the error analysis made, the MAPE from January 2009 to December 2018 of the first-order FTS and secondorder FTS is 7.240 % and 3.627 %, respectively, while the MAPE changes to 7.164% and 3.657 if included the recently collected four-month data of 2019. The MAPE for January 2019 to Apr 2019 is 5.124% and 4.376% for the first-order FTS and second-order FTS respectively. These MAPE results indicate that the second-order FTS has the highest accuracy and the best performance if compared to the first-order FTS. Hence, it is proven that better prediction can be achieved by applying higher-order FTS.

Acknowledgments
We are grateful to Universiti Tun Hussein Onn Malaysia for financially supporting this work under TiER 1 2018 grant number H258.