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
766 Nuramirah Akrom and Zuhaimy Ismail
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].
EMD-DR models for forecasting electricity load demand 767
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.
768 Nuramirah Akrom and Zuhaimy Ismail
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)
EMD-DR models for forecasting electricity load demand 769
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)
770 Nuramirah Akrom and Zuhaimy Ismail
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].
EMD-DR models for forecasting electricity load demand 771
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
772 Nuramirah Akrom and Zuhaimy Ismail
EMD-DR models for forecasting electricity load demand 773
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
774 Nuramirah Akrom and Zuhaimy Ismail
(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
EMD-DR models for forecasting electricity load demand 775
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
776 Nuramirah Akrom and Zuhaimy Ismail
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
EMD-DR models for forecasting electricity load demand 777
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.
References
[1] C. Chatfield, The Analysis of Time Series: An Introduction, Sixth Edition,
Chapman & Hall, New York, 2004.
[2] J. W. Taylor, L. M. D. Menezes and P. E. McSharry, A comparison of
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Received: April 20, 2016; Published: July 25, 2016