Adaptive Optimized Pattern Extracting Algorithm for Forecasting Maximum Electrical Load Duration Using Random Sampling and Cumulative Slope Index

Abstract

Load forecasting techniques can be an essential method to save energy and shave peak loads in order to improve energy efficiency and maintain the stability of a power grid. To achieve this goal, machine learning-based approaches have been proposed recently. Before moving toward the long-term and ultimate solution such as machine learning, we propose a simple and efficient method to forecast electricity usage patterns and the duration of maximum electrical load using a small data set. The proposed algorithm can forecast maximum electrical load duration using random sampling and a cumulative slope index. To verify the algorithm, we utilized electricity data (from 2015.11 to 2016.12) obtained from a building with a constant lifestyle and electricity pattern. The performance of the algorithm was evaluated using electricity bills, the discharging condition of an energy storage system, and the cumulative slope index. It was found that the proposed algorithm could provide electricity cost savings of 0.62–2.28% compared with other, conventional electricity prediction techniques, such as the moving average method and exponential smoothing. In near future research, it is expected that this algorithm could be applied to electrical big data to handle real-time data processing.

1. Introduction

There is a growing demand for electrical power as industry and economies develop that can sometimes result in insufficient power. To solve this problem, we are striving to expand the electricity supply facilities to enhance the power supply. However, despite our efforts, there is a limit to the extent to which supply facilities can be increased. Therefore, demand response (DR) management [1,2,3,4] of electrical power is a current trend and an alternative to focusing on expanding supply facilities. DR can provide a predictive method to control the power flow of an energy storage system (ESS), photovoltaic (PV) supply, buildings and grid [1] and based on a combination of a time-based and incentive-based DR program, a real-time incentive DR program can reduce the peak load [2]. DR programs using a game model [3,4] are effective in improving grid efficiency and reliability. The main policy of demand response is to improve energy efficiency and power load management. Regarding energy efficiency, forecasting electricity usage can be an essential method to save energy and shave peak loads. Research regarding load forecasting is currently in progress using statistical methods and time-series models.

Specifically, these load forecasting methods include regression analysis, exponential smoothing, and neural networks. Regression analysis [5,6,7,8] is a method used to predict the load and then utilize the data to forecast load and independent variables such as temperature and GDP (Gross Domestic Product) [5]. Exponential smoothing models [9,10,11,12] offer an alternative method to forecast electricity usage using recently measured data and different types of weighting. This method can improve the error rate of predicted data because the recent data has a greater value in the weighting. On the other hand, older data has weight that decreases exponentially over time. Neural networks are one method to artificially model the human brain mathematically and such models can require more and different types of weights and variables. Neural networks can also be linked to non-linear models to distinguish between dependent variables and independent ones. The load forecasting method that combines soft computing and neural networks is the new approach called hybrid methodologies [13]. This model is different from traditional forecasting models such as statistical methods. Data mining framework can also be an intelligent analysis and prediction of energy consumption based on electricity data [14,15].

In this paper, we propose an algorithm for forecasting the duration of maximum electrical load using random sampling and a cumulative slope index. The reasons that we choose a proposed approach are because we do not need a lot of training data to forecast electricity usage patterns. Second, only electricity data (single variable) can be utilized to predict maximum electrical load durations. Finally, the proposed approach is a simple and efficient method to maintain prediction accuracy without a complicated model before moving toward a long-term and ultimate solution such as machine learning. What we propose to do here is to forecast maximum electrical load duration for a building with a constant lifestyle and electricity pattern on a weekly basis (except for holidays and weekends: Saturdays and Sundays). To verify the algorithm, we utilized the electricity data (2015.11–2016.12) obtained from the headquarters building of Korea Electric Power Knowledge Data & Network (KEPCO KDN) located in Naju City, South Korea. The reason that the proposed algorithm can forecast the future electrical load of buildings is because household or office buildings have relatively constant electrical usage patterns. Therefore, for these, we can forecast the pattern of electrical use and maximum electrical load duration using the algorithm. Normally, we might use conventional and sequential correlation analysis to forecast the pattern and load duration. However, the proposed algorithm is intended to forecast electrical use patterns and maximum electrical load duration using repetitive correlation analysis with data extracted from random sampling to analyze as many data as possible to model the pattern of electrical usage.

By applying the proposed approach, we offer the potential that the electrical use pattern can be obtained more quickly than by using sequential correlation analysis, in terms of a stochastic method. In the second section, we will see how conventional and sequential correlation analysis is used to extract an electrical use pattern that can represent monthly data. In the third section, we propose the algorithm, defining an alternative approach to forecast electrical use patterns that can represent monthly data, employing random sampling and repetitive correlation analysis. Finally, to evaluate and flesh out the performance of the proposed algorithm, we will forecast maximum electrical load durations with each of three models (random, moving average, and exponential smoothing model). Then, we will discuss the quantitative results of the proposed algorithm using random sampling.

2. Sequential Correlation Analysis for Extracting Representative Electrical Pattern

2.1. Sequential Correlation Analysis

Correlation involves the analysis of statistical and mathematical interrelations between components of large data sets. It can be utilized to analyze statistical interrelations between two probability variables in various fields of engineering. For example, we can study correlation of wind and hydro with electricity demand and prices [16] and develop a method to evaluate the characteristics of team communications based on social network analysis [17]. To measure the correlation and similarity between two probability variables, covariance can be utilized as follows:

$$\begin{gathered} Cov\left(X,Y\right)={\sigma }_{XY}=E\left\{\left(X-{\mu }_X\right)\left(Y-{\mu }_Y\right)\right\} \\ =\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left(X_i-{\mu }_X\right)\left(Y_i-{\mu }_Y\right)\} \\ =E\left(XY\right)-E\left(X\right)E\left(Y\right), \end{gathered}$$

(1)

The correlation coefficient can be defined as in the following equation:

$$\rho \left(X,Y\right)=\frac{Cov\left(X,Y\right)}{\sqrt{Var\left(X\right)}\sqrt{Var\left(Y\right)}}=\frac{{\sigma }_{XY}}{{\sigma }_X{\sigma }_Y}$$

(2)

The values of the correlation coefficient occur within the range −1 to +1 and when the correlation shows a linear relation, the correlation value gets closer to +1. In contrast, when the correlation has a non-linear relationship, the value gets closer to −1. Generally, if the correlation coefficient values are 0.7 or higher, they have a strong linear relationship. When the values are between 0.3 and 0.7, the linearity is moderate, and with values below 0.3, the correlation values have a weak linear relationship.

Table 1. Statistical meanings of correlation coefficient values.

Correlation Value	Meaning
−1 to −0.7	Strong negative linear relationship
−0.7 to −0.3	Moderate negative linear relationship
−0.1 to +0.1	Negligible linear relationship
+0.1 to +0.3	Weak positive linear relationship
+0.3 to +0.7	Moderate positive linear relationship
+0.7 to +1.0	Strong positive linear relationship

shows the statistical meanings of correlation coefficients.

In this paper, we apply a correlation coefficient threshold of 0.8 (or 0.7) to decide the representative electrical pattern. This means that the electrical use pattern is defined as the pattern with correlation coefficient ≥0.7, which means a strong positive association between patterns.

2.2. Method for Extracting Representative Electrical Patterns for Each Month from the Original Data

In this section, we address the method used to extract the representative electrical pattern for each month from original data for the KEPCO KDN headquarters. The representative electrical pattern means the pattern of electrical usage of a specific date each month that could represent usage during the month. The main reason to extract the representative electrical pattern is to examine the performance of forecasting maximum electrical load duration for each model (random, moving average, and exponential smoothing model) compared to a representative electrical pattern. If the electrical pattern of a specific date within each month has probabilistically obvious affinities with the patterns of other dates (correlation coefficient ≥ correlation coefficient threshold) and also is in the majority within each month, the pattern can be a representative electrical pattern of the month. In this paper, a representative electrical pattern is defined as the pattern with correlation coefficient threshold of 0.8 that accounts for ≥65% (ratio threshold) within the data for a month. However, if any pattern within a month does not satisfy that condition, we apply the second definition of representative electrical pattern with correlation coefficient threshold of 0.7 that accounts for ≥65% within the data for a month. To achieve a representative electrical pattern, we perform correlation analysis sequentially using pre-processed data extracted from the original electricity data.

The Algorithm 1 explains how to extract the representative electrical pattern using sequential correlation analysis. As shown in Algorithm 1, to achieve a representative electrical pattern, we perform correlation analysis sequentially using pre-processed data extracted from the original electricity data. At first the correlation coefficient threshold was set at the correlation coefficient of 0.8, and then the ratio threshold was set at 65%. Second, we performed correlation analysis between components of two data sets sequentially. Finally, the pattern with the highest ratio among electrical patterns with a correlation coefficient ≥0.8 that accounts for ≥65% was extracted. As already mentioned, if any pattern within a month does not satisfy that condition, we apply the second definition of representative electrical pattern with correlation coefficient ≥0.7 that accounts for ≥65% within the data for a month, and then execute algorithm 1 once again.

Algorithm 1: The algorithm to extract the representative electrical pattern

1: $\mathrm{Cut}\_\mathrm{correl}\leftarrow \mathrm{Correlation} \mathrm{coefficient} \mathrm{threshold}$

2: $\mathrm{Cut}\_\mathrm{ratio}\leftarrow \mathrm{Ratio} \mathrm{threshold}$

3: $\mathrm{Correl}\left(\mathrm{i},\mathrm{j}\right)\leftarrow \mathrm{Statistical} \mathrm{correlation} \mathrm{function} \mathrm{between} \mathrm{i}-\mathrm{th} \mathrm{dataset} \mathrm{and} \mathrm{j}-\mathrm{th} \mathrm{dataset}$

4: $\mathrm{n}\leftarrow \mathrm{The} \mathrm{last} \mathrm{numberfield} \mathrm{among} \mathrm{dataset} \left\{{\mathrm{Date}}_{\left(1..\mathrm{n}\right)},\mathrm{Electrical}\_{\mathrm{Load}}_{\left(1..\mathrm{n}\right)}\right\}$

5: ${\mathrm{Record}}_{\mathrm{i}}=\left\{{\mathrm{T}}_{\mathrm{i}},{ \mathrm{R}}_{\mathrm{i}},{ \mathrm{Date}}_{\mathrm{i}}, \mathrm{Electrical}\_{\mathrm{Load}}_{\mathrm{i}}\right\}$

6: $\mathrm{B}\leftarrow {\mathrm{Memory} \mathrm{structure} \mathrm{to} \mathrm{store} \mathrm{Record}}_{\mathrm{i}}$

7: while $i\le n$ do

8: while $j\le n$ do

9: $C_{i,j}\leftarrow Correl\left(i,j\right)$

10: if $\left(C_{i,j}\ge Cut\_correl\right)$ then

11: $P_{i,j}\leftarrow 1$

12: else

13: $P_{i,j}\leftarrow 0$

14: end if

15: end while

16: $Rati{o}_i= \frac{{{\displaystyle \sum }}_{j=1}^n{P}_{i,j}}{n}$

17: if $\left({\mathrm{Ratio}}_{\mathrm{i}}\ge \mathrm{Cut}\_\mathrm{ratio}\right)$ then

18: $\mathrm{B}\leftarrow \mathrm{B}{\displaystyle \cup }{\mathrm{Record}}_{\mathrm{i}}$

19: end if

20: end while

21: $k\underset{p\in B}{\leftarrow argmax}\left(Recor{d}_p·T_p\right)$

22: $\mathrm{Extracted}\_\mathrm{Pattern}\leftarrow {\mathrm{Record}}_{\mathrm{k}}·\mathrm{Electrical}\_{\mathrm{Load}}_{\mathrm{k}}$

The concept of extracting representative electrical patterns and maximum electrical load duration using sequential correlation analysis is presented in Figure 1.

To forecast the duration of the maximum electrical load, we calculated cumulative slope index values for a representative electrical pattern and then, using the index, we compared it with each model for quantitative evaluation. The ratio of the representative electrical pattern with correlation coefficient ≥0.8 (or ≥0.7) is listed in

Table 2. Ratios of representative electrical patterns with correlation coefficient ≥0.8 (or ≥0.7).

Representative Electrical Pattern	Ratio (%) (Correlation Coefficient ≥ 0.8)	Ratio (%) (Correlation Coefficient ≥ 0.7)
2016.01	50	70
2016.02	48	67
2016.03	70	83
2016.04	86	90
2016.05	95	100
2016.06	91	95
2016.07	100	100
2016.08	96	96
2016.09	86	86
2016.10	76	86
2016.11	68	77
2016.12	73	82

. All these values were higher than 65% in case of correlation coefficient ≥0.8 (or ≥0.7) and represented the majority within each month. From these results, the use pattern of each month can be a representative of a statistical model of each month’s data in terms of stochastic method. As shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, the representative electrical patterns of a specific date within each month are different from each other. From these results, we could roughly know how the maximum electrical load durations of representative electrical patterns that we want to forecast are also different from each other.

3. Proposed Algorithm Using Random Sampling

In this section, we address the proposed algorithm using random sampling and repeated correlation analysis. Using the random sampling technique and repeated correlation analysis with previous data (such as one month ago or two months ago) within the period 2015.11–2016.12, we were able to find electricity usage patterns with strong correlation value in the electric data set. Actually, random sampling or Monte Carlo techniques [18,19,20,21] are widely used to solve various kinds of problems through the computation of random numbers. This research usually deals with uncertainty and the sensitivity of the energy model. These are also used to provide statistical estimates in various fields of engineering. These techniques can be applied to evaluate power systems’ composite reliability [18,19] and used for sensitivity analysis of scenario ranking for energy generation [20]. With the growing use of renewable energy resources, distributed generation (DG) systems are spreading and Monte Carlo techniques can address the probabilistic assessment of DG penetration to the distribution network [21]. Compared with sequential correlation analysis, the advantage of a correlation using random sampling is that it offers the potential to obtain electrical use patterns more quickly than with sequential correlation analysis, in terms of stochastic method. Using the proposed algorithm and random sampling, we might do less repetitive correlation analysis by using only the data extracted by random sampling. Algorithm 2 explains how to extract the pattern using random sampling. We perform correlation analysis between components of two data sets using random sampling to achieve a predicted pattern. At first, the correlation coefficient threshold was set at the correlation coefficient of 0.8, and then the ratio threshold was set at the specific value we want to achieve. The maximum repeated count was set at M values. Second, we performed correlation analysis between components of two data sets randomly and repeatedly. If the electrical pattern of a specific date was above the ratio threshold, the electrical use pattern for that sample date was recorded. Finally, when we encountered the maximum repeated count, the pattern among the sample dates with the highest ratio was extracted.

Algorithm 2: Proposed algorithm using random sampling

1: $\mathrm{Cut}\_\mathrm{correl}\leftarrow \mathrm{Correlation} \mathrm{coefficient} \mathrm{threshold}$

2: $\mathrm{Cut}\_\mathrm{ratio}\leftarrow \mathrm{Ratio} \mathrm{threshold}$

3: $\mathrm{Correl}\left(\mathrm{i},\mathrm{R}\right)\leftarrow \mathrm{Statistical} \mathrm{correlation} \mathrm{function} \mathrm{between} \mathrm{i}-\mathrm{th} \mathrm{dataset} \mathrm{and} \mathrm{R}-\mathrm{th} \mathrm{dataset}$

4: $\mathrm{M}\leftarrow \mathrm{Maximum} \mathrm{repeated} \mathrm{count}$

5:$\mathrm{n}\leftarrow \mathrm{The} \mathrm{last} \mathrm{numberfield} \mathrm{among} \mathrm{datasets} \left\{{\mathrm{Date}}_{\left(1..\mathrm{n}\right)},\mathrm{Electrical}\_{\mathrm{Load}}_{\left(1..\mathrm{n}\right)}\right\}$

6: ${\mathrm{Record}}_{\mathrm{i}}=\left\{{\mathrm{T}}_{\mathrm{i}},{ \mathrm{R}}_{\mathrm{i}},{ \mathrm{Date}}_{\mathrm{i}}, \mathrm{Electrical}\_{\mathrm{Load}}_{\mathrm{i}}\right\}$

7: $\mathrm{B}\leftarrow {\mathrm{Memory} \mathrm{structure} \mathrm{to} \mathrm{store} \mathrm{Record}}_{\mathrm{i}}$

8: while $\mathrm{count}\le \mathrm{M}$ do

9: $\mathrm{R}\leftarrow \mathrm{Random} \mathrm{Variable}\left(1..\mathrm{n}\right)$

10: while $\mathrm{i}\le \mathrm{n}$ do

11: ${\mathrm{C}}_{\mathrm{i},\mathrm{R}}\leftarrow \mathrm{Correl}\left(\mathrm{i},\mathrm{R}\right)$

12: if $({\mathrm{C}}_{\mathrm{i},\mathrm{R}}\ge \mathrm{Cut}\_\mathrm{correl})$ then

13: $P_{i,R}\leftarrow 1$

14: else

15: $P_{i,R}\leftarrow 0$

16: end if

17: end while

18: $Rati{o}_{count}= \frac{{{\displaystyle \sum }}_{i=1}^n{P}_{i,R}}{n}$

19: if $\left({\mathrm{Ratio}}_{\mathrm{count}}\ge \mathrm{Cut}\_\mathrm{ratio}\right)$ then

20: $\mathrm{B}\leftarrow \mathrm{B}{\displaystyle \cup }{\mathrm{Record}}_{\mathrm{count}}$

21: end if

22: $\mathrm{count}=\mathrm{count}+1$

23: end while

24: $\mathrm{k}\underset{\mathrm{j}\in \mathrm{B}}{\leftarrow \mathrm{argmax}}\left({\mathrm{Record}}_{\mathrm{j}}·{\mathrm{T}}_{\mathrm{j}}\right)$

25: $\mathrm{Extracted}\_\mathrm{Pattern}\leftarrow {\mathrm{Record}}_{\mathrm{k}}·\mathrm{Electrical}\_{\mathrm{Load}}_{\mathrm{k}}$

The concept of extracting the predicted pattern using random sampling is depicted in Figure 8.

To verify the appropriateness of the algorithm, we compared each representative electrical pattern with a predicted pattern extracted by the proposed algorithm. Comparisons of representative and predicted patterns are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. As shown in those figures, most of the predicted patterns bear a resemblance to the representative electrical pattern. From these results, we could roughly see the proposed algorithm can predict the maximum electrical load duration of the representative electrical pattern. However, the predicted patterns in Figure 14 cannot be seen as analogous to the representative electrical patterns. The cause of the problem might be that the link between the constant lifestyle and electrical load pattern are not the same as for the other patterns. In the next section, to evaluate the quantitative performance of the proposed algorithm using random sampling, we discuss the prediction results of maximum electrical load duration.

4. Performance Results of the Proposed Algorithm

In this section, we examine the performance of the proposed algorithm. For evaluation of its performance, we discuss the results of maximum electrical load duration. The reason that we attempted to find the maximum electrical load duration was to examine the electricity cost saving effects from use of the proposed algorithm. Maximized electricity cost saving effects occur when ESS are discharging electricity during the period of maximum electrical load. To confirm the duration of maximum electrical load, we propose using the concept of cumulative slope index, as well as the ESS discharging condition and KEPCO electricity bills, to evaluate a quantitative analysis. The flowchart for evaluating the performance is shown in Figure 15. As shown in Figure 15, we extracted pre-processed data from the original data and performed correlation analysis and then calculated the cumulative slope index values for each model. Finally, we compared them with the cumulative slope index value of representative electrical loads for quantitative evaluation.

4.1. The Concept of Cumulative Slope Index, Energy Storage System (ESS) Discharging Condition, and Korea Electric Power (KEPCO) Electricity Bill

4.1.1. The Concept of Cumulative Slope Index

To evaluate the algorithm quantitatively, the concept of the cumulative slope index can be used. The definition of the cumulative slope index, including other parameters, is shown in

Table 3. Parameter list of time, power, slope, cumulative slope and cumulative slope index.

Parameter
Time [h]	$T_0$	$T_1$	$T_2$	$\dots \dots$	$T_n$
Power of Predicted Pattern [kWh]	$P_0$	$P_1$	$P_2$	$\dots \dots$	$P_n$
Power of Representative Electrical Pattern [kWh]	$N_0$	$N_1$	$N_2$	$\dots \dots$	$N_n$
Slope of Predicted Pattern	$X_0$	$X_1$	$X_2$	$\dots \dots$	$X_{n-1}$
Slope of Representative Electrical Pattern	$S_0$	$S_1$	$S_2$	$\dots \dots$	$S_{n-1}$
Cumulative Slope of Predicted Pattern	$C{X}_0$	$C{X}_1$	$C{X}_2$	$\dots \dots$	$C{X}_{n-1}$
Cumulative Slope of Representative Electrical Pattern	$C{S}_0$	$C{S}_1$	$C{S}_2$	$\dots \dots$	$C{S}_{n-1}$
Cumulative Slope Index of Predicted Pattern [%]	$C{P}_0$	$C{P}_1$	$C{P}_2$	$\dots \dots$	$C{P}_{n-1}$
Cumulative Slope Index of Representative Electrical Pattern [%]	$C{R}_0$	$C{R}_1$	$C{R}_2$	$\dots \dots$	$C{R}_{n-1}$

.

The slope of the predicted pattern is determined as follows:

$$X_{n-1}=\frac{P_n-P_{n-1}}{T_n-T_{n-1}}$$

(3)

The slope of the representative electrical pattern is calculated as follows:

$$S_{n-1}=\frac{N_n-N_{n-1}}{T_n-T_{n-1}}$$

(4)

The cumulative slope index of the predicted pattern is calculated as follows:

$$\begin{array}{ccc}{T}_0 : X_0=C{X}_0& & C{P}_0=\frac{C{X}_0}{C{X}_{max}}\times 100[\%]\\ T_1 : X_0+X_1=C{X}_1& & C{P}_1=\frac{C{X}_1}{C{X}_{max}}\times 100[\%]\\ & ⋮& \\ & ⋮& \\ T_{n-1} : C{X}_{n-2}+X_{n-1}=C{X}_{n-1}& & C{P}_{n-1}=\frac{C{X}_{n-1}}{C{X}_{max}}\times 100[\%],\end{array}$$

(5)

The cumulative slope index of the representative electrical pattern is calculated as follows:

$$\begin{array}{ccc}{T}_0 : S_0=C{S}_0& & C{R}_0=\frac{C{S}_0}{C{S}_{max}}\times 100[\%]\\ T_1 : S_0+S_1=C{S}_1& & C{R}_1=\frac{C{S}_1}{C{X}_{max}}\times 100[\%]\\ & ⋮& \\ & ⋮& \\ T_{n-1} : C{S}_{n-2}+S_{n-1}=C{S}_{n-1}& & C{R}_{n-1}=\frac{C{S}_{n-1}}{C{S}_{max}}\times 100[\%],\end{array}$$

(6)

The slope ($\mathrm{X}, \mathrm{S} \mathrm{parameter}$) is defined as the electricity power variation divided by the time change. The cumulative slope of predicted pattern $(\mathrm{CX} \mathrm{parameter}$) and the cumulative slope of representative electrical pattern ($\mathrm{CS} \mathrm{parameter}$) are defined as the sum of the next slope value and the cumulative slope value. The cumulative slope index $\left(\mathrm{CP}, \mathrm{CR} \mathrm{parameter}\right)$ is defined as the ratio of the cumulative slope divided by the maximum cumulative slope (${\mathrm{CX}}_{\max },{ \mathrm{CS}}_{\max } \mathrm{parameter}$). For example, if the cumulative slope index is <80%, it means that the load is not within the maximum electrical load duration. In contrast, when the cumulative slope index is ≥80%, it indicates times within the maximum electrical load duration.

4.1.2. ESS Discharging Condition

Normally, an ESS cannot discharge continuously because of its limited capacity and lifecycle. Considering these characteristics, it is efficient to discharging an ESS only during the period of maximum electrical load. Therefore, the ESS is in discharging condition only while the cumulative slope index is over 80% (i.e., which means the electrical usage is 80% of maximum load). In this paper, the maximum electrical load duration is defined as being when the cumulative index is over 80%. When the ESS is discharging during the maximum electrical load, utility electricity is not used. At that time, we can provide electricity cost saving effects compared to using utility energy all the time. We evaluated the performance of the proposed algorithm under the ESS discharging condition that the cumulative slope index was over 80%.

4.1.3. KEPCO Electricity Bills

The KEPCO electricity bills were used to evaluate a quantitative analysis. The electricity costs from the bills are shown in

Table 4. Korea Electric Power (KEPCO) electricity bill of high voltage (A) type.

High Voltage (A)	Basic Chage (KRW/kW)	Time Zone	Electricity Usage Fare (KRW/kWh)
			Summer	Spring, Autumn	Winter
			June–August	March–May, Sep.–Oct.	Nov.–Feb.
	7220	Low load	61.6	61.6	68.6
		Medium load	114.5	84.1	114.7
		Maximum load	196.6	114.8	172.2

and

Table 5. KEPCO electricity bill for time zone and season.

Time Zone	Season
	Summer	Spring, Autumn	Winter
	June–August	March–May, Sep.–Oct.	Nov.–Feb.
Low load	23:00–09:00	23:00–09:00	23:00–09:00
Medium load	09:00–10:00	09:00–10:00	09:00–10:00
	12:00–13:00	12:00–13:00	12:00–17:00
	17:00–23:00	17:00–23:00	20:00–22:00
Maximum load	10:00–12:00 13:00–17:00	10:00–12:00 13:00–17:00	10:00–12:00 17:00–20:00 22:00–23:00

. We were able to perform quantitative evaluation by comparing the energy use predicted using the proposed algorithm against the predictions made with other techniques, using the actual costs from the bills.

4.2. Other Conventional Electricity Prediction Models (Moving Average Method and Exponential Smoothing)

Two conventional electricity prediction techniques are introduced here for comparison. One is the moving average method and the other is the exponential smoothing model.

The moving average is defined as follows:

$$F_t=\frac{1}{n}\left(D_{t-1}+D_{t-2}+\cdots +D_{t-n}\right)$$

(7)

As a sort of smoothing time series, this method is widely used to forecast the electricity usage because the model is simple and practical. By applying the model, we can extract the predicted pattern using the moving average method from average previous data (one and two months ago). The averaged previous data and the predicted pattern extracted from the averaged previous data using the moving average method are summarized in

Table 6. Average previous data and the predicted pattern extracted from averaged previous data using the moving average method.

Averaged Data for	Predicted Pattern for
2015.11–12	2016.01
2015.12–2016.01	2016.02
2016.01–02	2016.03
2016.02–03	2016.04
2016.03–04	2016.05
2016.04–05	2016.06
2016.05–06	2016.07
2016.06–07	2016.08
2016.07–08	2016.09
2016.08–09	2016.10
2016.09–10	2016.11
2016.10–11	2016.12

.

The exponential smoothing model is another sort of smoothing time series. However, there are some differences in the data size used, for forecasting using the moving average method and the exponential smoothing model. While the moving average method needs all the previous data (one and two months ago), the exponential smoothing model needs only a small amount of recent data.

In this paper, we applied data from the previous week (5 days) to forecast the monthly pattern. For example, to forecast the pattern of the month 2016.03, we applied the data from five days (02/23, 02/24, 02/25, 02/26, and 02/29) to forecast the pattern. The model is defined as follows:

$$F_t=\alpha X_{t-1}+\alpha \left(1-\alpha \right)X_{t-2}+\alpha {\left(1-\alpha \right)}^2{X}_{t-3}+\alpha {\left(1-\alpha \right)}^3{X}_{t-4}+\cdots +\alpha {\left(1-\alpha \right)}^n{X}_{t-n}$$

(8)

As you can see from the equation, the most recent data gets the largest weighting and the old data has weights that decrease exponentially over time. Each date having a different weight for forecasting a monthly pattern is listed in

Table 7. The predicted pattern list extracted from the exponential smoothing model using different weights of each date.

Previous Data					Predicted Pattern
Weight (α)					-
0.5	0.25	0.125	0.0625	0.03125	-
12/31	12/30	12/29	12/28	12/25	2016.01
1/29	1/28	1/27	1/26	1/25	2016.02
2/29	2/26	2/25	2/24	2/23	2016.03
3/31	3/30	3/29	3/28	3/25	2016.04
4/29	4/28	4/27	4/26	4/25	2016.05
5/31	5/30	5/27	5/26	5/25	2016.06
6/30	6/29	6/28	6/27	6/24	2016.07
7/29	7/28	7/27	7/26	7/25	2016.08
8/31	8/30	8/29	8/26	8/25	2016.09
9/30	9/29	9/28	9/27	9/26	2016.10
10/31	10/28	10/27	10/26	10/25	2016.11
11/30	11/29	11/28	11/25	11/24	2016.12

.

4.3. The Performance Results of the Proposed Algorithm

In this section, we examine the performance results of the proposed algorithm. To evaluate the performance, we address the ability to forecast the maximum electrical duration using the KEPCO electricity bills, ESS discharging condition, and cumulative slope index.

As already mentioned, the proposed algorithm performs correlation analysis between components of two data sets using random sampling to achieve a predicted pattern. Using the proposed algorithm, we might do less repetitive correlation analysis by using only the data extracted by random sampling. Then, we compare the performance results of the proposed algorithm with other, more conventional electricity prediction techniques such as moving average and exponential smoothing. To evaluate the performance of three models (random sampling, moving average, and exponential smoothing), the evaluation process was divided into two parts: forecasting accuracy and cost savings.

According to the predicted accuracy for quantitative evaluation, we could use cumulative slope index values. If the cumulative slope index of the representative electrical load was <80%, the load was not at maximum. In contrast, when the cumulative slope index of the representative electrical load was ≥80%, it means that conditions were within the maximum electrical load duration. The results of the cumulative slope index values for each model and representative electrical load are shown in

Table 8. Cumulative slope index values for three models (2015.11–12) and base model (2016.01).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	17–18	18–19	19–20	22–23
Random	CX	148.376	115.234	112.366	79.101	−7.812	−178.22	159.973
	CP	93%	72%	70%	49%	−5%	−111%	-
Moving Average	CX	240.631	219.31	209.799	128.534	52.983	−78.845	250.356
	CP	96%	88%	84%	51%	21%	−30%	-
Exponential Smoothing	CX	354.525	337.624	227.709	101.349	44.494	−55.237	354.525
	CP	100%	95%	64%	29%	13%	−16%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	17–18	18–19	19–20	22–23
Representative Electrical Load (2016.01)	CS	226.318	166.26	85.449	−24.902	−138.67	−255.37	288.696
	CR	78%	58%	30%	−9%	−48%	−88%	-

,

Table 9. Cumulative slope index values for three models (2015.12–2016.01) and base model (2016.02).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	17–18	18–19	19–20	22–23
Random	CX	382.568	385.925	217.346	152.832	121.582	−108.89	385.925
	CP	99%	100%	56%	40%	32%	−28%	-
Moving Average	CX	256.708	231.616	211.072	113.914	22.538	−129.79	256.708
	CP	100%	90%	82%	44%	9%	−51%	-
Exponential Smoothing	CX	125.557	95.304	52.869	−32.944	−127.27	−255.57	160.351
	CP	78%	59%	33%	−21%	−79%	−159%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	17–18	18–19	19–20	22–23
Representative Electrical Load (2016.02)	CS	407.166	365.845	296.265	262.695	124.512	59.082	481.018
	CR	85%	76%	62%	55%	26%	12%	–

,

Table 10. Cumulative slope index values for three models (2016.01–02) and base model (2016.03).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	347.717	353.638	346.558	330.566	320.007	316.833	578.064
	CP	60%	61%	60%	57%	55%	55%	-
Moving Average	CX	236.716	210.827	200.841	174.209	175.091	170.832	252.427
	CP	94%	84%	80%	69%	69%	68%	-
Exponential Smoothing	CX	297.642	278.204	274.313	254.744	222.309	246.555	299.379
	CP	99%	93%	92%	85%	74%	82%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.03)	CS	167.297	154.785	86.853	49.6826	64.5752	106.873	219.482
	CR	76%	71%	40%	23%	29%	49%	-

,

Table 11. Cumulative slope index values for three models (2016.02–03) and base model (2016.04).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	257.629	237.915	250.854	232.056	195.557	177.185	279.602
	CP	92%	85%	90%	83%	70%	63%	-
Moving Average	CX	245.565	219.137	214.525	183.964	190.482	178.725	284.44
	CP	86%	77%	75%	65%	67%	63%	-
Exponential Smoothing	CX	265.12	262.997	255.043	249.754	249.271	219.193	332.859
	CP	80%	79%	77%	75%	75%	66%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.04)	CS	237.427	217.224	226.074	219.421	250.977	216.187	250.97
	CR	95%	87%	90%	87%	100%	86%	-

,

Table 12. Cumulative slope index values for three models (2016.03–04) and base model (2016.05).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	281.128	272.217	316.04	300.476	292.847	252.93	316.04
	CP	89%	86%	100%	95%	93%	80%	-
Moving Average	CX	258.72	237.866	247.406	258.707	222.408	217.676	283.673
	CP	91%	84%	87%	91%	78%	77%	-
Exponential Smoothing	CX	255.423	229.015	226.505	299.784	152.504	188.19	299.784
	CP	85%	76%	76%	100%	51%	63%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.05)	CS	323.486	338.135	340.698	348.389	350.586	332.397	395.386
	CR	82%	86%	86%	88%	89%	84%	-

,

Table 13. Cumulative slope index values for three models (2016.04–05) and base model (2016.06).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	256.287	253.845	249.451	221.497	223.511	204.285	256.287
	CP	100%	99%	97%	86%	87%	80%	-
Moving Average	CX	282.594	293.04	297.768	311.254	280.826	291.83	311.254
	CP	91%	94%	96%	100%	90%	94%	-
Exponential Smoothing	CX	380.074	404.259	459.457	424.423	424.355	431.255	459.457
	CP	83%	88%	100%	92%	92%	94%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.06)	CS	561.035	545.776	576.05	597.656	593.384	605.713	605.713
	CR	93%	90%	95%	99%	98%	100%	-

,

Table 14. Cumulative slope index values for three models (2016.05–06) and base model (2016.07).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	459.717	515.747	412.598	471.558	463.745	365.845	515.747
	CP	89%	100%	80%	91%	90%	71%	-
Moving Average	CX	363.464	378.898	382.369	345.584	347.651	338.853	382.369
	CP	95%	99%	100%	90%	91%	89%	-
Exponential Smoothing	CX	452.591	419.415	437.798	256.737	237.038	208.317	452.591
	CP	100%	93%	97%	57%	52%	46%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.07)	CS	989.136	887.939	992.31	813.354	723.511	579.712	992.31
	CR	100%	89%	100%	82%	73%	58%	-

,

Table 15. Cumulative slope index values for three models (2016.06–07) and base model (2016.08).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	524.78	524.902	519.287	336.304	401.489	344.116	528.687
	CP	99%	99%	98%	64%	76%	65%	-
Moving Average	CX	541.268	534.134	539.378	426.315	425.534	402.724	541.268
	CP	100%	99%	100%	79%	79%	74%	-
Exponential Smoothing	CX	493.343	500.793	484.108	304.779	312.439	309.628	500.793
	CP	99%	100%	97%	61%	62%	62%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.8)	CS	606.445	599.487	583.496	360.229	266.235	287.354	606.445
	CR	100%	99%	96%	59%	44%	47%	-

,

Table 16. Cumulative slope index values for three models (2016.07–08) and base model (2016.09).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	524.78	524.902	519.287	336.304	401.489	344.116	528.687
	CP	99%	99%	98%	64%	76%	65%	-
Moving Average	CX	611.166	611.149	602.876	434.516	438.192	430.363	611.166
	CP	100%	100%	99%	71%	72%	70%	-
Exponential Smoothing	CX	503.796	530.418	449.402	467.888	499.29	516.468	530.418
	CP	95%	100%	85%	88%	94%	97%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.09)	CS	396.606	491.089	451.05	185.303	192.139	434.326	491.089
	CR	81%	100%	92%	38%	39%	88%	-

,

Table 17. Cumulative slope index values for three models (2016.08–09) and base model (2016.10).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	13–14	14–15	15–16	16–17
Random	CX	730.591	786.133	731.445	560.059	620.85	575.317	786.133
	CP	93%	100%	93%	71%	79%	73%	-
Moving Average	CX	483.42	510.541	503.518	325.418	313.352	393.655	510.541
	CP	95%	100%	99%	64%	61%	77%	-
Exponential Smoothing	CX	323.009	293.598	239.353	2.27737	22.0299	254.623	323.009
	CP	100%	91%	74%	1%	7%	79%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	13–14	14–15	15–16	16–17
Representative Electrical Load (2016.10)	CS	258.301	310.913	314.941	128.052	100.464	286.621	314.941
	CR	82.0%	99%	100%	41%	32%	91%	-

,

Table 18. Cumulative slope index values for three models (2016.09–10) and base model (2016.11).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	17–18	18–19	19–20	22–23
Random	CX	300.781	361.572	273.682	193.359	142.578	16.6016	380.981
	CP	79%	95%	72%	51%	37%	4%	-
Moving Average	CX	314.993	349.317	284.361	133.429	88.095	−34.77	352.096
	CP	89%	99%	81%	38%	25%	−10%	-
Exponential Smoothing	CX	202.793	204.102	168.503	138.824	88.1653	−4.6082	239.338
	CP	85%	85%	70%	58%	37%	−2%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	17–18	18–19	19–20	22–23
Representative Electrical Load (2016.11)	CS	62.6221	−122.19	−192.38	−191.41	−95.703	−226.56	94.1162
	CR	67%	−130%	−204%	−203%	−102%	−241%	-

and

Table 19. Cumulative slope index values for three models (2016.10–11) and base model (2016.12).

Model Group	Cumulative Slope Index	Maximum Load Time Zone						CXmax
		10–11	11–12	17–18	18–19	19–20	22–23
Random	CX	300.781	361.572	273.682	193.359	142.578	16.6016	380.981
	CP	79%	95%	72%	51%	37%	4%	-
Moving Average	CX	242.006	151.376	99.0814	35.0455	72.8334	−46.012	262.741
	CP	92%	58%	38%	13%	28%	−18%	-
Exponential Smoothing	CX	333.668	96.817	38.6772	−12.482	83.4656	−82.977	376.965
	CP	89%	26%	10%	−3%	22%	−22%	-
Base Model	Cumulative Slope Index	Maximum Load Time Zone						CSmax
		10–11	11–12	17–18	18–19	19–20	22–23
Representative Electrical Load (2016.12)	CS	202.881	−42.236	−213.87	−297.85	−217.77	−265.63	236.938
	CR	86%	−18%	−90%	−126%	−92%	−112%	-

.

As shown in Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18 and Table 19, some values of $\mathrm{CP} \mathrm{parameter}$ of cumulative slope index show that the predicted result of the method is analogous to the result of the representative electrical load or indicate that the method forecasts the maximum electrical load duration (cumulative slope index ≥80%) exactly while the representative electrical load was within the maximum electrical load duration.

In contrast, other values of $\mathrm{CP} \mathrm{parameter}$ of cumulative slope index show that the method forecasts the opposite result compared to the representative electrical load.

Through analysis of the results in Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18 and Table 19, the cumulative slope index values for the random sampling technique show 86% rates analogous to the representative electrical load among 72 maximum load time zones. The moving average shows 69% and the exponential smoothing model shows 67%.

These results show that the proposed algorithm using random sampling has a more analogous representative electrical pattern for obtaining probabilistically obvious affinities with the patterns for other dates (correlation coefficient ≥0.8) than do the other models (moving average and exponential smoothing).

According to the cost saving effects from quantitative evaluation, we performed an evaluation using the KEPCO electricity bill.

Table 20. Example for quantitative evaluation (2016.04) of three models.

Model Group	Cumulative Slope Index	Maximum Load Time Zone
		10–11	11–12	13–14	14–15	15–16	16–17
Random	92%	85%	90%	83%	70%	63%
Moving Average	86%	77%	75%	65%	67%	63%
Exponential Smoothing	80%	79%	77%	75%	75%	66%
Representative Electrical Load	95%	87%	90%	87%	100%	86%
Representative Electrical Load [kWh]		720.336	700.134	708.984	702.331	733.886	699.096
Unit Price (KEPCO) [KRW/kWh]	Model Group	Maximum Load Time Zone
		10–11	11–12	13–14	14–15	15–16	16–17
	Random	61.6	61.6	61.6	61.6	114.8	114.8
	Moving Average	61.6	114.8	114.8	114.8	114.8	114.8
	Exponential Smoothing	61.6	114.8	114.8	114.8	114.8	114.8
Fee (1 day) [KRW]	Model Group	Maximum Load Time Zone
		10–11	11–12	13–14	14–15	15–16	16–17
	Random	44,372.8	44,372.8	43,673.4	43,263.6	84,250.2	80,256.3
	Moving Average	44,372.8	80,375.4	81,391.4	80,627.7	84,250.2	80,256.3
	Exponential Smoothing	44,372.8	80,375.4	81,391.4	80,627.7	84,250.2	80,256.3

presents one quantitative evaluation and shows the evaluation result for 2016.04 for three models (random sampling, moving average, and exponential smoothing).

In the same way, the fee for the total duration (from 2016.01 to 2016.12: 12 months), can also be calculated. The evaluation results of 12-day fees and 240-day fees (weekdays over 12 months, excluding holidays) are shown in

Table 21. Cost savings and cost saving effects for quantitative evaluation of three models.

Fee	Model	Costs(KRW)	Costs(USD)
Total fee (12 days)	Random Sampling (RS)	19,544,636	18,477
	Moving Average (MA)	19,666,534	18,562
	Exponential Smoothing (ES)	19,990,568	18,867
Total fee (240 days)	Random Sampling (RS)	390,892,717	368,940
	Moving Average (MA)	393,330,672	371,241
	Exponential Smoothing (ES)	399,811,369	377,358
Cost savings between MA and RS (240 days)		2,437,955	2,301
Cost savings between ES and RS (240 days)		8,918,651	8,417
Cost Saving Effects
Cost savings (%) based on RS	Compared with MA	0.62%
	Compared with ES	2.28%

. As shown in Table 21, it was found that the proposed algorithm using random sampling could provide electricity cost savings of 0.62–2.28% greater than with the conventional electricity prediction techniques such as moving average and exponential smoothing for the total duration. Cost savings can be specified as the cost difference between moving average and random sampling, or between exponential smoothing and random sampling. Cost saving effects can be presented as cost saving percentages based on random sampling. When calculating the total electricity bill, we did not include basic charges and only considered the unit price in various time zones. This is because we only wanted to discuss the quantitative performance and efficiency of the proposed algorithm using random sampling.

5. Conclusions

In this paper, we propose an algorithm for forecasting maximum electrical load duration using random sampling and a cumulative slope index. Before moving toward the long-term and ultimate solution such as machine learning, we propose a simple and efficient method to forecast electricity usage patterns and the duration of maximum electrical load using a small data set. The algorithm can forecast electricity usage patterns and maximum electrical load duration for a building with a constant lifestyle and electricity use pattern on a weekly basis utilizing a random sampling technique.

In contrast to sequential correlation analysis, for which we might examine as many data as possible to determine the pattern of electrical usage, the proposed algorithm does repetitive correlation analysis with data extracted by random sampling. With the proposed approach, this paper offers the potential for electrical use patterns to be obtained more quickly than with sequential correlation analysis, in terms of stochastic method.

To verify the effectiveness of the algorithm, we used electricity data (2015.11–2016.12) for the headquarters building of KEPCO KDN located in Naju City, South Korea, to forecast the electrical use pattern. We proposed an algorithm to provide an alternative approach to forecasting the electrical pattern to represent the usage data each month, employing random sampling and repetitive correlation analysis. To evaluate the performance of the proposed algorithm, we forecast the duration of maximum electrical load with each model (random sampling, moving average, and exponential smoothing) and the quantitative performance of the algorithm was evaluated using KEPCO electricity bills, ESS discharging condition, and cumulative slope index. As a result, it was found that the proposed algorithm can provide electricity cost savings of 0.62–2.28% greater than with other, more conventional electricity prediction techniques such as moving average and exponential smoothing. In near-future research, it is expected that this algorithm will become applicable to electrical applications involving big data, to deal with real-time data processing. It should also offer an alternative method for exploring the use of ESS systems for incorporation into various complex big data systems.