research article a hybrid short-term power load...

Research ArticleA Hybrid Short-Term Power Load Forecasting Model Based onthe Singular Spectrum Analysis and Autoregressive Model

Hongze Li Liuyang Cui and Sen Guo

School of Economics and Management North China Electric Power University Beijing 102206 China

Correspondence should be addressed to Sen Guo guosen324163com

Received 15 February 2014 Revised 17 April 2014 Accepted 17 April 2014 Published 5 May 2014

Academic Editor Mamun B I Reaz

Copyright copy 2014 Hongze Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Short-term power load forecasting is one of the most important issues in the economic and reliable operation of electricitypower system Taking the characteristics of randomness tendency and periodicity of short-term power load into account a newmethod (SSA-AR model) which combines the univariate singular spectrum analysis and autoregressive model is proposed Firstlythe singular spectrum analysis (SSA) is employed to decompose and reconstruct the original power load series Secondly theautoregressive (AR) model is used to forecast based on the reconstructed power load seriesThe employed data is the hourly powerload series of the Mid-Atlantic region in PJM electricity market Empirical analysis result shows that compared with the singleautoregressive model (AR) SSA-based linear recurrent method (SSA-LRF) and BPNN (backpropagation neural network) modelthe proposed SSA-AR method has a better performance in terms of short-term power load forecasting

1 Introduction

Short-term power load forecasting is one of the most impor-tant issues in economic and reliable operation of powersystem Many operating decisions related to electricity powersystem such as unit commitment dispatch scheduling of thegenerating capacity reliability analysis security assessmentand maintenance scheduling of the generators are based onthe short-term power load forecasting

In recent years domestic and foreign scholars have donemany studies in the field of short-term power load forecast-ing Currently the short-term power load forecastingmethodcan be divided into two categories that is load-series-basedforecasting method and affecting-factors-based forecastingmethod Although the power load shows the random anduncertain characteristic it also has an apparent tendencyTherefore the load-series-based forecasting method is basedupon the internal structure of the short-term power loadseries which includes ARIMA ARMAX [1 2] neural net-works [3 4] gray prediction model [5 6] wavelet analysis [78] and other forecasting methods However these methodshave some shortcomings the load-series-based forecastingmethod can only be used for data fitting and is not suitablefor the treatment of regularity the neural network method

has the problem that the relation between the input variablescannot be expressed explicitly the grey prediction model isused for the case of the little sample data the wavelet analysisforecasting method transforms the original sequence by theorthogonal wavelets to get the subsequences of differentfrequency-domain and then uses the subsequences to predictand reconstruct which has a higher accuracy than using theoriginal series directlyThe affecting-factors-based method isfrom the perspective of the factors affecting the power loadwhich uses the regression analysis to perform the predictionon the base of determining the relation between differentvariables [9] However it is very difficult to find all thefactors accurately and comprehensively which are influencedby the geographical factors economic situation and climateand other factors that are often quite different in differentperiods to some extent So it is too hard to find an equationwhich can be suited to all the forecasting cases Althoughthe affecting-factors-based method is better than the load-series-based in theory it has much less practical operabilityMoreover the affecting-factors-based method can be putinto use for forecasting the medium- and long-term powerload To our knowledge and related work [10 11] the key ofshort-term power load forecasting is to grasp the primaryingredients reflecting the variation tendency and it is quite

Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2014 Article ID 424781 7 pageshttpdxdoiorg1011552014424781

2 Advances in Electrical Engineering

necessary to find out a method that can depict the fluctuationcharacteristics of power load series

The singular spectrum analysis (SSA) technology is a typ-ical time-series-based analysis method which has been usedfor industrial production forecasting [12] signals detection[13] electricity price forecasting [14] murmur detection fromheart sounds [15] and so on The aim of SSA is to make adecomposition of the original time series into the sum of asmall number of independent and interpretable componentssuch as a slowly varying trend oscillatory components anda structureless noise Then some of these components areused for time series forecasting At present the SSA methodhas been widely applied to cope with the problem in manydomains such as geography and sociology [16 17]

From the perspective of the fluctuation characteristicsthe power load shows randomness trend and periodicitywhich can be extracted by using the SSA method That isto say the stochastic noise components which influence theforecasting accuracy can be eliminated And Afshar andBriceno have already applied this method to load forecastingwith linear recurrent formulae (LRF) [18 19] but as we allknow the relation of power load in different time is notsimply linear usually complicated nonlinear relationship ispresented So the SSA-LRF is not so perfect if we take thisinto consideration Meanwhile using the time-series-basedmodel to forecast power load can not only overcome theproblem of invalid linear fitting but also make up for theshortage of failing to handle the regularity of time seriesTherefore in this paper the SSA approach and autoregressive(AR) model are combined to forecast the short-term powerload that is SSA-AR forecasting model

The rest of this paper is organized as follows In Section 2the SSAmethodology and theARmodel are described brieflyand the hybrid SSA-AR power load forecasting model isintroduced the empirical analysis is performed in Section 3and the forecasting results of several different forecastingmodels are presented and discussed Section 4 concludes thispaper

2 The Basic Principle of SSA-AR Model

21 A Brief Introduction to SSA Method Singular spectrumanalysis (SSA) method contains two phases named decom-position and reconstruction The former phase arranges theoriginal sequence in a form of time-delay matrix beforedecomposing the original time series Then the time seriesare reconstructed via grouping anddiagonal averagingwhichis called reconstructionThe reconstructed series is then usedfor forecasting the new data points

211 Decomposition Given a time series119884 = (1198841 1198842 119884

119879)

119879 gt 2 In the first step the embedding procedure is appliedto construct a sequence of 119883

119894= (119910119894minus1 119910

119894+119871minus2)119879 with 119871-

dimension vectors from the time series 119884119879

119883 =[[

[

1199101sdot sdot sdot 119910119870

d

119910119871 119910119879

]]

]

(1)

where 119883 is a Hankel matrix with equal elements on thediagonals 119894 + 119895 = const119870 = 119879 minus 119871 + 1

In the second step the 119871 times 119871 matrix 119883119883119879 is calculatedand its Eigen triples (119904

119894119880119894 and119881

119894) are determined by singular

value decomposition (SVD) Denote the Eigen values of119883119883119879by 120582119894(119894 = 1 2 119871) in descending order and let119880

119894and119881

119894be

the 119894th left and right Eigen vectors of119883119883119879 respectively Set 119889= rank(119883) Then the trajectory matrix119883 can be rewritten as

119883 = 1198831+ 1198832+ sdot sdot sdot + 119883

119889

119883119894= radic120582

119894119880119894119881119894

(2)

where 119904119894is the 119894th singular value of119883 and119883

119894(119894 = 1 119889) are

the matrices of rank one

212 Reconstruction After decomposition the time seriesis reconstructed via grouping and diagonal averaging Ingrouping step the indices 119869 = 119894 = 1 119889 are grouped into119872 disjoint subsets 119868

1 119868

119872corresponding to splitting the

elementary matrices 119883119894(119894 = 1 119889) into 119872 groups Each

group contains a set of indices as 119868 = 1198941 119894

119901 Then the

resultant matrix119883119894is defined as119883

119868119894= 1198831198941+1198831198942+ sdot sdot sdot +119883

119894119901 so

119883 = 1198831198681+ 1198831198682+ sdot sdot sdot + 119883

119868119872 (3)

where the trajectory matrix 119883 is represented as a sum of119872 resultant matrices Next the diagonal averaging transferseach matrix119883

119868119899(119899 = 1 119872) into a time series

Let 119883 be a (119871 times 119870) matrix with elements 119909119894119895 Set 119871lowast =

min(119871 119870)119870lowast = max(119871 119870) and

119884lowast=

119884 119871 le 119870

119884119879 119871 gt 119870

(4)

where 119884lowast isin 119877119871lowasttimes119870lowast

Diagonal averaging transfers the matrix119883 to a series 119866 =

(1198920 119892

119879minus1) according to the following

119892119896=

1

119896 + 1

119896+1

sum

119898=1

119909lowast

119898119896minus119898+2 0 le 119896 lt 119871

lowastminus 1

1

119871lowast

119871lowast

sum

119898=1

119909lowast

119898119896minus119898+2 119871

lowastminus 1 le 119896 lt 119870

lowast

1

119879 minus 119896

119879minus119896+1

sum

119898=119896minus119896lowast+2

119909lowast

119898119896minus119898+2 119870lowastminus 1 le 119896 lt 119879

(5)

22 A Brief Introduction to AR Model The principle ofautoregressive (AR) model is to use the current interferenceand the limited past observations to predict the present valueGiven a time series 119884 = (119910

1 1199102 119910

119905) the mathematical

representation of the 119901-order autoregressive model is asfollows

119910119905= 1205721119910119905minus1+ 1205722119910119905minus2+ sdot sdot sdot + 120572

119901119910119905minus119901+ 120576119905 (6)

where 120572119894(119894 = 1 2 119901) is the undetermined coefficients of

the model (also called autoregressive coefficient) and 120576119905is a

Advances in Electrical Engineering 3

Original series

Establish a trajectory matrix

Singular value decomposition

Reconstruct theseries by diagonal

averaging

ADF unit root test

The series is stationary

Construct an AR model

Forecast

p-order difference of the series

Yes

No

The series is p-order stationary

SSA

AR

Figure 1 The specific flow chart of the SSA-AR model

random disturbance whose mean value is zero but variance isnot equal to zero

If the lag operator is defined as 120593 then 120593119910119905= 119910119905minus1 120593119896119910119905=

119910119905minus119896

The 119901-order autoregressive model can be presented asfollows

119910119905= (1205721120593 + 12057221205932+ sdot sdot sdot + 120572

119901120593119901) 119910119905+ 120576119905≜ Φ (120593) 119910

119905+ 120576119905 (7)

Only the reciprocal of all roots of lag operatorrsquos polyno-mial is less than one (both fall within the unit circle) the AR(119901) process is called covariance stationary process

23 Introduction to SSA-AR Model The application processof SSA-AR model for short-term power load forecasting ismainly divided into two steps first using the SSA methodto decompose and reconstruct the original power loadseries and then using the AR model to forecast with thereconstructed sequences The specific calculation procedureof SSA-AR model for short-term power load forecasting isshown in Figure 1

3 Empirical Analysis

The hourly power load series of Mid-Atlantic region in thePJM electricitymarket from 18 June to 18 July 2013 containing720 sample points is employed in the experimental data Thementioned data versus time have been shown in Figure 2

31 Decomposing and Reconstructing the Original Power LoadSeries Based on SSA As mentioned earlier the window

20

25

30

35

40

45

50

55

60

65

1 109 217 325 433 541 649 757

Load

Hour

times103

Figure 2 The hourly load curve of the Mid-Atlantic region fromJune 18 to July 18 in 2013

0

1000

2000

3000

4000

5000

6000

7000

1 10 19 28 37 46 55 64 73 82

Sing

ular

val

ue

Order

times103

Figure 3 Singular values curve (in descending order)

length 119871 is the only parameter in the decomposition stage Ifthe time series has a periodic component the window lengthis taken proportional to that period to get better separabilityTherefore 119871 = 24 times 7 = 168 h is assumed here which corre-sponds to weekly variations of power load time series Thiswindow length results in 168 Eigen triples Then the singularvalue decomposition (SVD) is applied to the time-delaymatrix by MATLAB programming Figure 3 illustrates thetrend of the 168 singular point values

As shown in Figure 3 the convergence rate of the singularvalue is very fast The first 30 singular values have a share of996of the power load serieswhich cannearly be consideredas the weekly trend The singular values from the 30thone are basically close to zero which comprise the randomcomponent of the original power load serials Therefore theoriginal serials should be omitted when reconstructed Thefirst 30 singular values in descending order (taken to the base-10 logarithm) are listed in Table 1

The decomposition helps us to make the proper groupsto extract the trend the harmonic component and random


Table 1 The first 30 singular values in descending order (taken tothe base-10 logarithm)

Order log(119909) Order log(119909) Order log(119909)1 1631 11 1206 21 11132 1415 12 1187 22 11073 1415 13 1187 23 11074 1339 14 1173 24 10805 1303 15 1167 25 10766 1249 16 1148 26 10687 1223 17 1143 27 10618 1216 18 1136 28 10619 1211 19 1126 29 106010 1208 20 1113 30 1060

minus1500

minus1000

minus500

0

500

1000

1500

1 55 109 163 217 271 325 379 433 487 541 595 649 703Load

Hour

Figure 4The gap between the reconstructed power load series (RS)and the original power load series

noise component Based on the relevantmathematical theoryabout the singular value decomposition there are three basicconclusions (1) the first Eigen triple almost represents thewhole trend of the time series (2) every harmonic componentwith a different frequency produces two Eigen triples withclose singular values (3) as a rule a pure noise series producesa slowly decreasing sequence of singular values the explicitplateau in the Eigen value spectra is caused by two Eigentriples with close singular values As shown in Table 1 itis obvious that the Eigen triples of (2 3) (12 13) (22 23)(27 28) and (29 30) produce the main harmonics of powerload series

Based on the above analysis this paper tries to reconstructthe power load series using the first 30 Eigen values Figure 4shows the gap between the original power load series (OS)Just as shown in Figure 4 the reconstruction has been donewith a satisfactory error

Figure 4 reflects that the reconstructed power load seriesfit well with the original series of which the correlationcoefficient reaches 09988 Therefore the decomposition andreconstruction of the original power load series not onlygreatly reduce the matrix dimension but also extract themain ingredient of the original series So we can describe itsdynamic change trend in a better way and also get a betterpredicting outcome

minus15minus1

minus050

051

15

1 3

AC

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

PAC

Figure 5 The autocorrelation and partial correlation of the recon-structed series

Table 2 The result of ADF test on first order difference power loadseries

Item ADF statistics ProbOriginal series minus277525 00623First order difference minus372147 00040

Table 3 The regression result of AR (3) model of reconstructedpower load series

Variable Coefficient Std error 119905-statistic ProbAR (1) 2067776 0035534 5819094 00000AR (2) minus1396917 0066505 minus2100462 00000AR (3) 0263791 0035598 7410290 00000

Parament ValueAdjusted 119877-squared 0986002Schwarz criterion 1356932Durbin-Watson stat 1967281

32 Forecasting the Reconstructed Power Load Series Basedon AR Model Because the AR model is only applicable tothe stationary time series the first thing that needs to bedone is to examine the stationarity of the reconstructed powerload series with the ADF unit root test By employing thesoftware Eviews 60 we can get the 119875 value equal to 00623 gt005 which means that the null hypothesis should not berejected That is to say the reconstructed power load seriesis a nonstationary time series Then consider the stationarityof its first order difference series and the calculation result islisted in Table 2 It can be seen that the first order differenceseries is stationary

Observe the autocorrelation coefficient and partial corre-lation coefficient of the first order difference sequence whichare shown in Figure 5 we can see the autocorrelation coef-ficient gradually decreases with fluctuation and the partialcorrelation coefficient tends to zero after the third orderTherefore the AR (3) model can be established which isshown as follows

Δ119910119905= 120572Δ119910

119905minus1+ 120573Δ119910

119905minus2+ 120574Δ119910

119905minus3+ 120576119905 (8)

Then estimate (8) with the ordinary least squares (OLS)method and the regression result is listed in Table 3


00

05

AR

root

s

10

15Inverse roots of ARMA polynomial(s)

minus15

minus10

minus05

00minus15 minus10 minus05 05 10 15

Figure 6 Unit circle test of the covariance stationarity

FromFigure 6 we can see that the reciprocals of two rootsof lag operatorrsquos polynomial of AR (3) both fall within theunit circle which indicates that this process is covariancestationary

Then expand the sample size to 745 from 744 and use theAR (3) model to forecast its value By calculation the 745thfirst-order differential value equals minus3620273 Therefore the745th point value can be forecasted which equals 42867027In the same way the following value at any point of a certainperiod can also be forecasted

33 Comparing the Forecasting Results of Different ModelsThe AR model SSA-LRF model and BPNN model areselected as the comparative models The ARmodel is appliedto the original power load series without any treatmentthat can extract the main trend Although AR model canwell represent the whole tendency of the original seriesthe predicted value is not perfect in a way The SSA-LRFmodel is the combination of SSAmethod and linear recurrentformula (LRF) in which LRF is a simple linear combinationof the known data and its coefficients are determined bySSA method BPNN (backpropagation neural network) ismade of neuron Not only are sufficient neurons connectedto net properly but the BPNN model should be trainedappropriately before it can simulate all types of nonlinearcharacteristics BPNN model is applied most widespreadamong all artificial neural network models which has beenwidely applied to many fields related to forecasting Thepredicted hourly power load results in 24 hours of Mid-Atlantic region in PJM market on July 19 by employing theabove four forecasting methods are depicted in Figure 7

In order tomeasure the performance of the three forecast-ing methods two indices that is hourly mean error (HME)and the hourly peak error (HPE) have been employed in

35

45

55

65

1 3 5 7 9 11 13 15 17 19 21 23

Load

Hour

ActualARSSA-AR

SSA-LRFBPNN

times103

Figure 7 The comparison of prediction results of different modelsand the actual value

this paper The HME and HPE are the well-known statisticalindices for evaluating prediction methods defined as follows

HME = 1

24

24

sum

119894=1

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

HPE = Max1le119894le24

(

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

)

(9)

where 119871119894ACT and 119871

119894FOR are the actual and predicted powerload of hour 119894 respectively

The calculation results of absolute error rate HME andHPE are shown in Figure 8 It can be seen that the value ofSSA-AR model in terms of HME and HPE is 058 and 257respectively and the absolute error rate of SSA-AR model isthe smallest compared with the other three models Theseindicate that the SSA-AR model shows better performancethan SSA-LRF AR and BPNN model in terms of short-termpower load forecasting

4 Conclusions

In this paper a hybrid short-term power load forecastingmodel based on the singular spectrum analysis (SSA) andautoregressive (AR) model is proposed As we all know theshort-term power load forecasting is vital in the fundamentaloperational functions of electricity market such as unitcommitment economic dispatch interchange evaluationscheduled maintenance and security assessment In thispaper the SSA-AR model has been employed as a tool forshort-term power load forecasting Firstly the power loadseries is analyzed with the SSAmethod to obtain the effectiveand predictable components of power load series Thenthe AR method is used to forecast the future values of thepower load seriesThe hybrid SSA-AR power load forecasting


0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900

AR SSA-AR SSA-LRF BPNN

Abso

lute

erro

r rat

e (

)

DMEDPE

(b)

Figure 8 The forecasting error comparison of four methods (a)absolute error rate (b) HME and HPE

method is examined by using the experimental data of Mid-Atlantic region inPJMelectricitymarketTheobtained resultsshow that the proposed method has a good ability in theprediction of the desired power load series However there isone point that needs emphasis this method does not take thefactors influencing the power load fluctuation into accountOnce the outside situation such as political economic andclimate condition has a sudden change this methodmay notwork Therefore this method is applicable to the short-termpower load forecasting without the tremendous changes ofoutside situation

Conflict of Interests

The authors declare that they have no conflict of interests

Acknowledgment

This study is supported by the Beijing Philosophy and SocialScience Planning Project (no 11JGB070)

References

[1] H A Amarawickrama and L C Hunt ldquoElectricity demand forSri Lanka a time series analysisrdquo Energy vol 33 no 5 pp 724ndash739 2008

[2] S S Pappas L Ekonomou D C Karamousantas G E Chatza-rakis S K Katsikas and P Liatsis ldquoElectricity demand loadsmodeling using AutoRegressive Moving Average (ARMA)modelsrdquo Energy vol 33 no 9 pp 1353ndash1360 2008

[3] D-X Niu H-F Shi andDDWu ldquoShort-term load forecastingusing bayesian neural networks learned byHybridMonte Carloalgorithmrdquo Applied Soft Computing Journal vol 12 no 6 pp1822ndash1827 2012

[4] D-M Zhou X-H Guan J Sun and Y Huang ldquoA short-termload forecasting system based on BP artificial neural networkrdquoPower System Technology vol 26 pp 10ndash14 2002

[5] G-D Li C-H Wang S Masuda and M Nagai ldquoA researchon short term load forecasting problem applying improved greydynamic modelrdquo International Journal of Electrical Power ampEnergy Systems vol 33 no 4 pp 809ndash816 2011

[6] D-M Shi L-C Li and J-W Song ldquoPower system load fore-casting based upon combination of gray forecast and artificialneural networkrdquo Power System Technology vol 25 pp 13ndash162001

[7] G Jie ldquoApplication of wavelet analysis to short-term loadforecasting of power systemrdquo Proceedings of the EPSA vol 15pp 40ndash45 2003

[8] H M Al-Hamadi and S A Soliman ldquoLong-termmid-termelectric load forecasting based on short-term correlation andannual growthrdquo Electric Power Systems Research vol 74 no 3pp 353ndash361 2005

[9] P Peng and J-H Peng ldquoResearch on the prediction of powerload based on multiple linear regression modelrdquo Journal ofSafety Science and Technology vol 7 pp 158ndash161 2011

[10] H-Z Li S Guo C-J Li and J-Q Sun ldquoA hybrid annual powerload forecasting model based on generalized regression neuralnetwork with fruit fly optimization algorithmrdquo Knowledge-Based Systems vol 37 pp 378ndash387 2013

[11] H Li S Guo H Zhao C Su and B Wang ldquoAnnual electricload forecasting by a least squares support vector machine witha fruit fly optimization algorithmrdquo Energies vol 5 no 11 pp4430ndash4445 2012

[12] H Hassani S Heravi and A Zhigljavsky ldquoForecasting UKindustrial productionwithmultivariate singular spectrumanal-ysisrdquo Journal of Forecasting vol 32 no 5 pp 395ndash408 2013

[13] M Rui C Yushu and S Huagang ldquoDetection of weak signalsbased on empiricalmode decomposition and singular spectrumanalysisrdquo IET Signal Processing vol 7 no 4 pp 269ndash276 2013

[14] A Miranian M Abdollahzade and H Hassani ldquoDay-aheadelectricity price analysis and forecasting by singular spectrumanalysisrdquo IETGeneration TransmissionampDistribution vol 7 no4 pp 337ndash346 2013

[15] S Sanei M Ghodsi and H Hassani ldquoAn adaptive singularspectrum analysis approach to murmur detection from heartsoundsrdquoMedical Engineering amp Physics vol 33 no 3 pp 362ndash367 2011

[16] L Telesca T Matcharasvili T Chelidze and N ZhukovaldquoRelationship between seismicity and water level in the Engurihigh dam area (Georgia) using the singular spectrum analysisrdquoNatural Hazards and Earth System Sciences vol 12 pp 2479ndash2485 2012


[17] H Hassani A S Soofi and A A Zhigljavsky ldquoPredictingdaily exchange rate with singular spectrum analysisrdquo NonlinearAnalysis Real World Applications vol 11 no 3 pp 2023ndash20342010

[18] K Afshar and N Bigdeli ldquoData analysis and short term loadforecasting in Iran electricity market using singular spectralanalysis (SSA)rdquo Energy vol 36 no 5 pp 2620ndash2627 2011

[19] H Briceno C M Rocco and E Zio ldquoSingular spectrumanalysis for forecasting of electric load demandrdquo ChemicalEngineer Transactions vol 33 pp 919ndash924 2013

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necessary to find out a method that can depict the fluctuationcharacteristics of power load series

The singular spectrum analysis (SSA) technology is a typ-ical time-series-based analysis method which has been usedfor industrial production forecasting [12] signals detection[13] electricity price forecasting [14] murmur detection fromheart sounds [15] and so on The aim of SSA is to make adecomposition of the original time series into the sum of asmall number of independent and interpretable componentssuch as a slowly varying trend oscillatory components anda structureless noise Then some of these components areused for time series forecasting At present the SSA methodhas been widely applied to cope with the problem in manydomains such as geography and sociology [16 17]

From the perspective of the fluctuation characteristicsthe power load shows randomness trend and periodicitywhich can be extracted by using the SSA method That isto say the stochastic noise components which influence theforecasting accuracy can be eliminated And Afshar andBriceno have already applied this method to load forecastingwith linear recurrent formulae (LRF) [18 19] but as we allknow the relation of power load in different time is notsimply linear usually complicated nonlinear relationship ispresented So the SSA-LRF is not so perfect if we take thisinto consideration Meanwhile using the time-series-basedmodel to forecast power load can not only overcome theproblem of invalid linear fitting but also make up for theshortage of failing to handle the regularity of time seriesTherefore in this paper the SSA approach and autoregressive(AR) model are combined to forecast the short-term powerload that is SSA-AR forecasting model

The rest of this paper is organized as follows In Section 2the SSAmethodology and theARmodel are described brieflyand the hybrid SSA-AR power load forecasting model isintroduced the empirical analysis is performed in Section 3and the forecasting results of several different forecastingmodels are presented and discussed Section 4 concludes thispaper

2 The Basic Principle of SSA-AR Model

21 A Brief Introduction to SSA Method Singular spectrumanalysis (SSA) method contains two phases named decom-position and reconstruction The former phase arranges theoriginal sequence in a form of time-delay matrix beforedecomposing the original time series Then the time seriesare reconstructed via grouping anddiagonal averagingwhichis called reconstructionThe reconstructed series is then usedfor forecasting the new data points

211 Decomposition Given a time series119884 = (1198841 1198842 119884

119879)

119879 gt 2 In the first step the embedding procedure is appliedto construct a sequence of 119883

119894= (119910119894minus1 119910

119894+119871minus2)119879 with 119871-

dimension vectors from the time series 119884119879

119883 =[[

[

1199101sdot sdot sdot 119910119870

d

119910119871 119910119879

]]

]

(1)

where 119883 is a Hankel matrix with equal elements on thediagonals 119894 + 119895 = const119870 = 119879 minus 119871 + 1

In the second step the 119871 times 119871 matrix 119883119883119879 is calculatedand its Eigen triples (119904

119894119880119894 and119881

119894) are determined by singular

value decomposition (SVD) Denote the Eigen values of119883119883119879by 120582119894(119894 = 1 2 119871) in descending order and let119880

119894and119881

119894be

the 119894th left and right Eigen vectors of119883119883119879 respectively Set 119889= rank(119883) Then the trajectory matrix119883 can be rewritten as

119883 = 1198831+ 1198832+ sdot sdot sdot + 119883

119889

119883119894= radic120582

119894119880119894119881119894

(2)

where 119904119894is the 119894th singular value of119883 and119883

119894(119894 = 1 119889) are

the matrices of rank one

212 Reconstruction After decomposition the time seriesis reconstructed via grouping and diagonal averaging Ingrouping step the indices 119869 = 119894 = 1 119889 are grouped into119872 disjoint subsets 119868

1 119868

119872corresponding to splitting the

elementary matrices 119883119894(119894 = 1 119889) into 119872 groups Each

group contains a set of indices as 119868 = 1198941 119894

119901 Then the

resultant matrix119883119894is defined as119883

119868119894= 1198831198941+1198831198942+ sdot sdot sdot +119883

119894119901 so

119883 = 1198831198681+ 1198831198682+ sdot sdot sdot + 119883

119868119872 (3)

where the trajectory matrix 119883 is represented as a sum of119872 resultant matrices Next the diagonal averaging transferseach matrix119883

119868119899(119899 = 1 119872) into a time series

Let 119883 be a (119871 times 119870) matrix with elements 119909119894119895 Set 119871lowast =

min(119871 119870)119870lowast = max(119871 119870) and

119884lowast=

119884 119871 le 119870

119884119879 119871 gt 119870

(4)

where 119884lowast isin 119877119871lowasttimes119870lowast

Diagonal averaging transfers the matrix119883 to a series 119866 =

(1198920 119892

119879minus1) according to the following

119892119896=

1

119896 + 1

119896+1

sum

119898=1

119909lowast

119898119896minus119898+2 0 le 119896 lt 119871

lowastminus 1

1

119871lowast

119871lowast

sum

119898=1

119909lowast

119898119896minus119898+2 119871

lowastminus 1 le 119896 lt 119870

lowast

1

119879 minus 119896

119879minus119896+1

sum

119898=119896minus119896lowast+2

119909lowast

119898119896minus119898+2 119870lowastminus 1 le 119896 lt 119879

(5)

22 A Brief Introduction to AR Model The principle ofautoregressive (AR) model is to use the current interferenceand the limited past observations to predict the present valueGiven a time series 119884 = (119910

1 1199102 119910

119905) the mathematical

representation of the 119901-order autoregressive model is asfollows

119910119905= 1205721119910119905minus1+ 1205722119910119905minus2+ sdot sdot sdot + 120572

119901119910119905minus119901+ 120576119905 (6)

where 120572119894(119894 = 1 2 119901) is the undetermined coefficients of

the model (also called autoregressive coefficient) and 120576119905is a


Original series




averaging

ADF unit root test



Forecast


Yes

No


SSA

AR




119910119905minus119896


119910119905= (1205721120593 + 12057221205932+ sdot sdot sdot + 120572

119901120593119901) 119910119905+ 120576119905≜ Φ (120593) 119910

119905+ 120576119905 (7)






20

25

30

35

40

45

50

55

60

65

1 109 217 325 433 541 649 757

Load

Hour

times103


0

1000

2000

3000

4000

5000

6000

7000

1 10 19 28 37 46 55 64 73 82

Sing

ular

val

ue

Order

times103








minus1500

minus1000

minus500

0

500

1000

1500

1 55 109 163 217 271 325 379 433 487 541 595 649 703Load

Hour





minus15minus1

minus050

051

15

1 3

AC

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

PAC









Δ119910119905= 120572Δ119910

119905minus1+ 120573Δ119910

119905minus2+ 120574Δ119910

119905minus3+ 120576119905 (8)



00

05

AR

root

s

10


minus15

minus10

minus05







35

45

55

65

1 3 5 7 9 11 13 15 17 19 21 23

Load

Hour

ActualARSSA-AR

SSA-LRFBPNN

times103



HME = 1

24

24

sum

119894=1

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT


(

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

)

(9)

where 119871119894ACT and 119871



4 Conclusions



0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900


Abso

lute

erro

r rat

e (

)

DMEDPE

(b)





Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Original series




averaging

ADF unit root test



Forecast


Yes

No


SSA

AR




119910119905minus119896


119910119905= (1205721120593 + 12057221205932+ sdot sdot sdot + 120572

119901120593119901) 119910119905+ 120576119905≜ Φ (120593) 119910

119905+ 120576119905 (7)






20

25

30

35

40

45

50

55

60

65

1 109 217 325 433 541 649 757

Load

Hour

times103


0

1000

2000

3000

4000

5000

6000

7000

1 10 19 28 37 46 55 64 73 82

Sing

ular

val

ue

Order

times103








minus1500

minus1000

minus500

0

500

1000

1500

1 55 109 163 217 271 325 379 433 487 541 595 649 703Load

Hour





minus15minus1

minus050

051

15

1 3

AC

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

PAC









Δ119910119905= 120572Δ119910

119905minus1+ 120573Δ119910

119905minus2+ 120574Δ119910

119905minus3+ 120576119905 (8)



00

05

AR

root

s

10


minus15

minus10

minus05







35

45

55

65

1 3 5 7 9 11 13 15 17 19 21 23

Load

Hour

ActualARSSA-AR

SSA-LRFBPNN

times103



HME = 1

24

24

sum

119894=1

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT


(

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

)

(9)

where 119871119894ACT and 119871



4 Conclusions



0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900


Abso

lute

erro

r rat

e (

)

DMEDPE

(b)





Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












minus1500

minus1000

minus500

0

500

1000

1500

1 55 109 163 217 271 325 379 433 487 541 595 649 703Load

Hour





minus15minus1

minus050

051

15

1 3

AC

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

PAC









Δ119910119905= 120572Δ119910

119905minus1+ 120573Δ119910

119905minus2+ 120574Δ119910

119905minus3+ 120576119905 (8)



00

05

AR

root

s

10


minus15

minus10

minus05







35

45

55

65

1 3 5 7 9 11 13 15 17 19 21 23

Load

Hour

ActualARSSA-AR

SSA-LRFBPNN

times103



HME = 1

24

24

sum

119894=1

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT


(

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

)

(9)

where 119871119894ACT and 119871



4 Conclusions



0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900


Abso

lute

erro

r rat

e (

)

DMEDPE

(b)





Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

05

AR

root

s

10


minus15

minus10

minus05







35

45

55

65

1 3 5 7 9 11 13 15 17 19 21 23

Load

Hour

ActualARSSA-AR

SSA-LRFBPNN

times103



HME = 1

24

24

sum

119894=1

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT


(

1003816100381610038161003816119871 119894ACT minus 119871 119894FOR1003816100381610038161003816

119871119894ACT

)

(9)

where 119871119894ACT and 119871



4 Conclusions



0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900


Abso

lute

erro

r rat

e (

)

DMEDPE

(b)





Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

002001

004003

006005

008007

009

1 3 5 7 9 11 13 15 17 19 21 23

Abso

lute

erro

r rat

e

Hour

AR-SSAAR

AR-LRFBPNN

(a)

000100200300400500600700800900


Abso

lute

erro

r rat

e (

)

DMEDPE

(b)





Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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