emd-dr models for forecasting electricity load demand · emd-dr models for forecasting electricity...

Contemporary Engineering Sciences, Vol. 9, 2016, no. 16, 763 - 780

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ces.2016.6430

EMD-DR Models for Forecasting

Electricity Load Demand

Nuramirah Akrom and Zuhaimy Ismail

Department of Mathematical Sciences

Faculty of Sciences

Universiti Teknologi Malaysia

81310, Skudai, Johor, Malaysia

Copyright © 2016 Nuramirah Akrom and Zuhaimy Ismail. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

Forecasting electricity demand is a vital process since electricity is a hard-to-store

resource. To accurately forecast electricity demand, this paper proposes a novel

method combining Empirical Mode Decomposition (EMD) and Dynamic

Regression namely EMD-DR method. EMD is a technique for detecting

non-stationary and nonlinear signal, while Dynamic Regression approach is a

method that involves lagged external variables. The EMD-DR method was

applied to a half-hourly of electricity demand (kW) and reactive power (var) of

Malaysia; where the reactive power data act as exogenous variable for Dynamic

Regression method. This paper demonstrates that the proposed EMD-DR model

provides a better forecast compared to a single Dynamic Regression model.

Keywords: Empirical Mode Decomposition, Dynamic Regression, Interpolation,

Reactive Power

1 Introduction

Load forecasting is essential to the current power system operation,

application, and regulation. As lead times differ, various load forecasts are desired

for different objectives. Short-term load forecasting with lead times ranging from

one day to a few days is important to operation planning, safety consideration,

preservation scheduling, and project set-up for both power propagation and

distribution facilities. Consequently, increasing the accuracy of short-term load

764 Nuramirah Akrom and Zuhaimy Ismail

forecasts can improve the sustainability of power supply organization and

demand.

In the last decades, various methods have been put forward for load

forecasting. The methods can be categorized as; i) univariate method; ii)

multivariate method; and iii) combined method [1]. Univariate methods include

exponential smoothing [2], Box-Jenkins approach [3], nonparametric functional

methods [4], Kalman filters [5], Artificial neural network (ANN) [6] and Support

Vector Machine (SVM) [7] are linear and non-linear models. Multivariate

methods, that include lagged external variables such as Multivariate Adaptive

Regression Splines (MARS) [8], GARCH method [9], Multivariate

Non-parametric Regression [9] and Nonlinear Autoregressive models with

exogenous inputs (NAX) [10] have produced great results for short term load

forecasting. Most researchers used temperature [11] and [12], wind generation

[13], special day effects [14] as their exogenous variables in multivariate methods.

Unfortunately, no single method, either univariate or multivariate has achieved

satisfactory results for short-term load forecasting. In consequence, combined

methods were created and successfully implemented for short-term load

forecasting [15] and [16].

In recent years, methods that combined Empirical Mode Decomposition

(EMD) with other techniques as proposed by An et al. [17], Dong et al. [18] and

Fan et al. [19] were used in short-term load forecasting. EMD is an adaptive

signal decomposition technique that uses the Hilbert-Huang Transform (HHT)

and can be applied to non-linear and non-stationary time series [20]. EMD is

based on straightforward presumption that any signal contains distinct intrinsic

modes of oscillations. Each nonlinear or non-stationary mode has equal number of

zero-crossings and extrema. There is only one extremum amongst consecutive

zero-crossings. Each mode must be independent of the others. The aim of the

EMD method is to decompose a signal into a number of Intrinsic Mode Functions

(IMFs) [20].

The combination of EMD with Dynamic Regression method gives a novel

approach for short-term load forecasting. In this paper, Dynamic Regression

model acts as a benchmark for measuring EMD-DR, where reactive power acts as

exogenous variable in Dynamic Regression method. EMD was used to decompose

the input (electricity load demand) and output (reactive power) to its distinct IMFs

and residue separately and fit suitable DR models in the decompose series. Finally,

the prediction results obtained from different EMD-DR models were aggregated.

The rest of the paper proceeds as follows. The next section explains the data

series, followed by the methodology. The Results section describes some results

and discussion and lastly conclusion section.

EMD-DR models for forecasting electricity load demand 765

2 The Data Series

In this work, we examined two types of data, which are electricity load

demand and reactive power data.

Electricity Load Demand Data

Figure 1: The time series plot of electricity demand and reactive power from

January 1, 2013 to May 31, 2013

A five-month Malaysia half hourly load demand and reactive power series

from January 1, 2013 to May 31, 2013 were used in this study. Both data for

electricity load demand (kW) and reactive power (kvar) were plotted in Figure 1

and it consists of 7248 observations (20 weeks). The data were provided by the

Malaysian electricity utility company, Tenaga Nasional Berhad (TNB). Figure 1

indicates that the daily cycles and the pattern of data electricity load demand and

reactive power are mostly identical. We observed that the cycles for Monday (7

January 2013) through Friday (11 January 2013) are similar, whereas the cycles

for weekends are quite distinct. There is a sharp decrease from the pattern of

data especially during the public holidays, which were during the Chinese New

Year (10 February 2013 to 12 February 2013). From the plot, we noted that the

electricity load demand increases from 18 February 2013 to 31 May 2013. This

clearly shows that the customers’ usage increase and therefore, it is necessary to


forecast short term half-hourly electricity load demand for an efficient operational

planning of utility companies. The half-hourly utility demand data and reactive

power data exhibit both daily and weekly cycles. Hence, double seasonality type

method was applied in this paper.

Reactive Power Data

Reactive power is the amplitude of power oscillation with no net transfer of

energy and it is caused by energy storage components, such as capacitor or inductor [21]. Although it does not contribute to the transfer of energy, it loads the

equipment as if it does utilize active power. Reactive power is also available in a

process involving reactive (capacitive or inductive) elements and can be either

constructed or utilized by distinct load or production components. Even though

“imaginary”, reactive power has substantial physical importance and it is highly

significant to the distribution planning of the electrical components [22],[23].

Therefore, we used reactive power as an exogenous variable because reactive

power plays an important role in distribution and transmission planning of

electricity demand and it acts as a leading indicator in the benchmark model.

3 Methodology

Double Seasonal Dynamic Regression Method

Transfer Function Model

The general form of Transfer Function Model is as follows [24]:

ttt nxBvy )( (1)

where tx and ty are assumed to be properly transformed and stationary series

while for a single-predictor, single-dependent linear system, the dependent

variable ty and the predictor input tx are related through a linear filter in

equation (1).

i

j

j BvBv )( is attributed to the transfer function of sifter, and tn is the noise

series of the system that is independent of the input series, tx .

When tx and tn are assumed to follow some ARIMA models, Equation

(1),which is also known as the ARIMAX model where “X” stands for exogenous

variable. Pankratz [24] called the Transfer Function Models as Dynamic

Regression model. Dynamic Regression (DR) model shows how a dependent

variable, tY linearly corresponds to past and current values of one or more

independent variables, tntt XXX ,,2,1 ,,, [24].


Double Seasonal ARIMA Model

Double seasonal periods exist in the electricity load demand and reactive

power data, which are weekly and daily seasonal, hence double seasonal

multiplicative ARIMA model was implemented. The multiplicative double

seasonal ARIMA model is expressed as [25]:

t

S

Q

s

Qqt

DS

DSdS

P

S

Pp

aBBBZB

BBBBB

)()()()1(

)1()1)(()()(

2

2

1

1

22

112

2

1

1

(2)

where tZ approximately transforms electricity load demand in period t; B

denotes backshift operator; )(Bp and )(Bq are regular autoregressive and

moving average polynomials of orders p and q; ),(),(),( 1

1

2

2

1

1

S

Q

S

P

S

P BBB

and )( 2

2

S

Q B are moving average and autoregressive polynomials of orders,

121 ,, QPP and 2Q ; 1S and 2S are seasonal periods; d, 1D and 2D are the orders of

integration; ta is a white noise process with zero mean and constant variance.

The seasonal cycles 1S and 2S are associated to the type of load data and reactive

power data series.

Dynamic Regression Method

Equation (3) shows the rational form of a model with M inputs and M

transfer functions. When tN is referred to autocorrelated function, acted as

disturbance series where tt aN and follow ARIMA model, it is a DR model

[24]:

tti

M

i i

b

i

t NXB

BBCY

i

,

1 )(

)(

(3)

where C is the constant term. Hence, the complete DR model is

td

p

q

ti

M

i i

b

it a

BB

BX

B

BBCY

i

)1)((

)(

)(

)(,

1

(4)

when the series involves seasonality, SARIMA model must be used for noise

series where it is an extended of double seasonal ARIMA model when the series

have double seasonality.


Empirical Mode Decomposition (EMD) Method

The EMD approach is used to decompose a non-stationary and non-linear

signal into Intrinsic Mode Functions (IMFs) [20]. Any signal )(tx can be

decomposed into IMFs. The decomposition process can be summarized in five

steps:

1) Identify all the local extrema. Then, connect all the local maxima, which are

the upper envelope, to local minima, which are the lower envelope, by using a

cubic spline method.

2) Design the mean of upper and lower envelopes as 1m ,

3) Set the difference between the signal )(tx and 1m as the first component

1h .

11)( hmtx (5)

Ideally, if 1h is an IMF, then 1h is the first component )(tx i.e,

4) If 1h is not an IMF, treat 1h as the pioneer signal and repeat step 1, 2 and 3.

Hence,

11111 hmh (6)

where 11m refers to the mean of lower and upper envelope values of 1h .

After repeated filtering, i.e. up to k times, kh ,1 turns into an IMF, which is:

kkk hmh 11)1(1 (7)

Hence, let khc 11 the first IMF element from the pioneer data 1c should consist

of the best scale or the shortest period element of the signal.

5) Isolate 1c from )(tx .Then, obtain the following equation:

11 )( ctxr (8)

Treat 1r as the pioneer data and reiterate the above process. This yields the

second IMF element 2c of )(tx . Repeat the process as described above n times,

to get the n-IMFs of signal )(tx . Hence,

nnn rcr

rcr

1

221

(9)


Stop the decomposition when nr becomes a monotonic function from which no

more IMF can be extracted. Sum up equation (8) and (9). Then the final equation

n

j

nj rctx1

)( (10)

Residue nr denotes the mean trend of )(tx . The IMFs 𝑐1, 𝑐2, ⋯ , 𝑐𝑛 include

distinct frequency bands varying from high frequency to low frequency. The

frequency elements involve in individual frequency band are distinct and they

interchanged with the deviation of signal )(tx , while nr represents the central

tendency of signal )(tx .

Interpolation

Interpolation is a method of constructing new data points within the range of

a discrete set of known data points. The moving average interpolation method is a

method that assigns values to a series by averaging the data within the series and

adds the neighboring target series. To use moving average, the data series must be

identified and the minimum number of data to use is specified. The average

equation is as follows:

)(11

1

1

n

n

i

i xxn

xn

x

(11)

where x is the mean of the series, n is the numbers given, each number denoted by

ix , where ni ,,1 The average is the sum of ix ’s divided by n. The new series *

1,tX and *

5,tX can be constructed by adding neighboring (the series before (*

1,tX )

and after (*

5,tX )) averaging series. Consider the following series:

,,6,5,4,31 X

,,4,3,2,12 X

,,7,6,2,23 X

,,7,9,5,84 X

,,8,2,3,75 X

The new series *

1,tX can be constructed by taking average of 32 , XX and 4X .

Then, adding it to 1X . This can be expressed as:

)(3

1432`11, XXXXX t

(12)


Hence, the new series of *

5,tX can be expressed as:

)(

3

1432`55, XXXXX t

(13)

The EMD-DR Combined Method

This paper proposes a method using EMD and DR to predict the electricity

load demand. The basic idea in implementing EMD-DR model is in using EMD to

decompose the input and output to its distinct IMFs and the residue separately and

then to fit suitable DR models to the decomposed series. Finally, the prediction

results obtained from the different EMD-DR models are aggregated. The steps to

achieve the final forecast can be summarized as follows:

1) The IMFs and residue component of the input and output data were extracted

separately using the sifting process. The number of IMFs and residue component

for all the output data must be the same as input data. If the number of IMFs for

the output data was not the same, the interpolation process is applied.

2) For each IMF and residue component obtained in Step 1, an appropriate

Dynamic Regression model, known as IMF-DR model, is developed.

3) The predictions obtained from the EMD-DR models in Step 2 were aggregated

together. This prediction equation is used to forecast the data series.

Mean Average Percentage Error (MAPE)

The Mean Absolute Percentage Error (MAPE) can be considered to estimate

the performance of a model prediction. The equation is as follows:

100|ˆ

ˆ|

1

1

n

t t

tt

F

FA

n (14)

where tA denotes the actual values, tF̂ denotes the forecasted values, and n

denotes the number of the forecasted values. Among the accuracy measures, the

MAPE is commonly used in forecasting literatures because MAPE expresses the

error in percentage value and easy for researcher to make a comparison for

forecast performance with other methods [26].


4 Results

IMFs for Electricity Load Demand and Reactive Power Data

The IMFs for electricity and reactive power data were obtained by using

EMD extraction algorithm as discussed in the Methodology section. 16 IMFs and

a residue (IMF 17) compound were obtained from the electricity load demand

data, while 19 IMFs and a residue (IMF 20) component were extracted from

reactive power data.

Figure 2 shows the IMFs and residue for electricity load demand (blue color)

and reactive power (green color). EMD gives the local characteristics, the

periodicity, the randomness and the trends of the original load and reactive power

[27]. It is observed from Figure 3 that the frequency of oscillation of the data is

declining as the IMFs are being extracted from the first through the last one. The

residue represents an indication of the extended period trend of the data series.

There are high frequencies of oscillation of IMF1 to IMF 8 (for both,

electricity load demand and reactive power) and also there are significantly

similar frequencies between electricity load demand and reactive power data.

Even though the frequencies are quite high, this is not a major drawback of

implementing predictive equations of these IMF, because the frequencies indicate

stationary data.

IMF9 to IMF 19 show that the frequencies are already stabilized with a

constant mean and variance. EMD decomposition process also shows the number

IMFs of electricity load demand and reactive power are not the same. This is due

to the fact that EMD has no specified “basis”. Its “basis” is adaptively produced

depending on the signal itself. The data characteristic of electricity load demand

and reactive power is not the same, leading to different numbers of IMFs between

them. The residue of IMFs indicates a long-term change of the mean for

electricity load demand and reactive power. The extraction of EMD yields

different numbers of IMFs and residue for electricity demand (input data) and

reactive power (output data).

Hence, interpolations were performed at IMFs 16, 17, 18, 19 and 20 of

reactive power series. The objective of implementing interpolation process is to

obtain the same number of IMFs and residue for input and output data. Figure 3

represents the new IMF 16 and IMF 17(residue) of reactive power after the

interpolation process.

IMF

2

652558005075435036252900217514507251

750000

500000

250000

0

-250000

-500000

IMF

2

652558005075435036252900217514507251

400000

300000

200000

100000

0

-100000

-200000

-300000

-400000

IMF

1

652558005075435036252900217514507251

300000

200000

100000

0

-100000

-200000

-300000

IMF

3

652558005075435036252900217514507251

3000000

2000000

1000000

0

-1000000

-2000000

-3000000

IMF

3

652558005075435036252900217514507251

500000

400000

300000

200000

100000

0

-100000

-200000

-300000

-400000

IMF

4

652558005075435036252900217514507251

2000000

1000000

0

-1000000

-2000000

IMF

4

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

5

652558005075435036252900217514507251

2000000

1000000

0

-1000000

-2000000

IMF

5

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

6

652558005075435036252900217514507251

2000000

1000000

0

-1000000

-2000000

IMF

6

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

7

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

7

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

IMF

1

652558005075435036252900217514507251

400000

300000

200000

100000

0

-100000

-200000

-300000

-400000



IMF

8

652558005075435036252900217514507251

1500000

1000000

500000

0

-500000

-1000000

-1500000

IMF

8

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

IMF

9

652558005075435036252900217514507251

1500000

1000000

500000

0

-500000

-1000000

IMF

9

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

-750000

IMF

10

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

10

652558005075435036252900217514507251

750000

500000

250000

0

-250000

-500000

IMF

11

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

11

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

12

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

12

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

-750000

IMF

13

652558005075435036252900217514507251

500000

0

-500000

-1000000

IMF

13

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

IMF

14

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

IMF

14

652558005075435036252900217514507251

500000

250000

0

-250000

-500000

IMF

18

652558005075435036252900217514507251

200000

100000

0

-100000

-200000

IMF

19

652558005075435036252900217514507251

300000

200000

100000

0

-100000

-200000

-300000

-400000

-500000

-600000

resid

ue

652558005075435036252900217514507251

3400000

3200000

3000000

2800000

2600000

2400000

2200000


(a) Electricity load demand (blue color) (b) reactive power (green color)

Figure 2: The IMFs and residue (red color) (IMF 1 to IMF17) of electricity load

demand (blue color) and the IMFs and residue (red color) (IMF 1 to IMF 20) of

reactive power (green color)

IMF

15

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

-1500000

IMF

15

652558005075435036252900217514507251

200000

100000

0

-100000

-200000

-300000

-400000

IMF

16

652558005075435036252900217514507251

1000000

500000

0

-500000

-1000000

-1500000

-2000000

-2500000

IMF

16

652558005075435036252900217514507251

150000

100000

50000

0

-50000

-100000

resid

ue

652558005075435036252900217514507251

13000000

12000000

11000000

10000000

9000000

IMF

17

652558005075435036252900217514507251

200000

100000

0

-100000

-200000


Figure 3: The new IMF 16 (purple color) and IMF 17(residue) of reactive power

after the interpolation process

Prediction Equation

The second step in building the EMD-DR combined method was conducted

after the IMFs and the residue components were obtained. During this step, the

enhancement to DR method was developed and called IMF-DR model. The final

16 IMFs and a residue (IMF 17) for electricity load demand predictive equations

are tabulated in Table 1.

Table 1: Electricity Load Demand Predictive Equations Using EMD-DR Model

IMFs Predictive equations

IMF1 12

32121

01420.007289.0

04741.04994.0053.1033.16434.09430.0

tt

ttttttt

aa

xxxxyyy

IMF2

2

166543

21321

4464.0

5181.005787.005787.01018.009991.009732.0

4701.09901.05770.02086.0263.1859.1

t

tttttt

ttttttt

a

axxxxx

xxxyyyy

IMF3 214

32121

7104.0207.11222.0

1256.01631.03788.02509.08749.0807.1

ttt

ttttttt

aax

xxxxyyy

IMF4 2121 2897.06571.03608.09660.0890.1 tttttt xxxyyy

IMF5 2

12121

8639.0

741.105733.01655.01094.09759.0958.1

t

ttttttt

a

axxxyyy

IMF6 21

32121

9125.0852.1

06901.009949.001125.004162.09850.0973.1

tt

ttttttt

aa

xxxxyyy

IMF7 tttt xyyy 0003092.09978.09993.1 21

IMF8 tttt xyyy 4

21 10518.89866.0985.1

Index

IMF

16

652558005075435036252900217514507251

750000

500000

250000

0

-250000

-500000

IMF

17

652558005075435036252900217514507251

3200000

3000000

2800000

2600000

2400000

2200000


Table 1: (Continued): Electricity Load Demand Predictive Equations Using

EMD-DR Model

IMF9 tttt xyyy 4

21 10388.29952.0999.1

IMF10 211 9996.0999.101517.09995.0 ttttt aaxyy

IMF11 21 9999.0202114.0 tttt aaxy

IMF12 tttt xyyy 6

21 10516.12

IMF13 tttt xyyy 7

21 10809.69998.02

IMF14 tttt xyyy 4

21 10487.39996.02

IMF15 tttt xyyy 5

21 10806.39999.02

IMF16 21

9

21 9999.0210073.82

tttttt aaxyyy

IMF17 ttt xyy 4

1 10071.69994.0

The last step in building EMD-DR model is by combining the seventeen

predictive equations to obtain the final predictive equation for the electricity load

demand as follows:

7̂1

6̂15̂14̂13̂12̂11̂10̂19̂

8̂7̂6̂5̂4̂3̂2̂1̂ˆ

IMF

IMFIMFIMFIMFIMFIMFIMFIMF

IMFIMFIMFIMFIMFIMFIMFIMFYt

(15)

Electricity Load Demand Forecasting using EMD-DR Model

The final predictive equation (15), was used to forecast the data series. The

forecasting performance was compared between EMD-DR model and traditional

Dynamic Regression model. This was to compare the efficiency of the EMD-DR

technique against the traditional technique. Table 2 shows the forecast accuracy

for electricity load demand using EMD-DR and Dynamic Regression models.

Table 2: Comparison of Forecasting Performance using EMD-DR and Dynamic

Regression Models

Forecasts

MAPE (%) Percentage

Improvement

(%) EMD-DR Model Dynamic

Regression Model

In-sample forecast 1.0674 1.1059 3.4813

Out-sample forecasts 0.7237 0.8074 10.3666


Table 2 shows the EMD-DR model has increased the forecast accuracy of the

classical Dynamic Regression model by 3.4813% and out-sample one-month

forecasts by 10.3666%, which are very significant. This shows that the EMD-DR

technique is a good method to forecast electricity load demand.

5 Conclusion

This paper presents a new technique for forecasting electricity load demand

using EMD-DR method. The proposed approach exploits the combined strength

of EMD-DR, which outperforms the forecast accuracy. There is an improvement

of forecasting accuracy by 3.4813% for in-sample forecast and 10.3666% for

out-sample forecast using the combined method EMD-DR, as compared with

traditional single model Dynamic Regression.

Acknowledgements. The authors would like to thank the Malaysian Ministry of

Higher Education for their My Master Scholarship and Universiti Teknologi

Malaysia for their financial allowance through Zamalah Scholarship and also to

Tenaga Nasional Berhad (TNB) Malaysia for providing the load data.

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Received: April 20, 2016; Published: July 25, 2016

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