research article intelligent optimized combined model...

Research ArticleIntelligent Optimized Combined Model Based onGARCH and SVM for Forecasting Electricity Price ofNew South Wales Australia

Yi Yang1 Yao Dong2 Yanhua Chen1 and Caihong Li1

1 School of Information Science and Engineering Lanzhou University Lanzhou 730000 China2 School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China

Correspondence should be addressed to Yao Dong dongsky2005gmailcom

Received 14 January 2014 Accepted 26 February 2014 Published 6 April 2014

Academic Editor Haiyan Lu

Copyright copy 2014 Yi Yang et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Daily electricity price forecasting plays an essential role in electrical power system operation and planning The accuracy offorecasting electricity price can ensure that consumers minimize their electricity costs and make producers maximize their profitsand avoid volatility However the fluctuation of electricity price depends on other commodities and there is a very complicatedrandomization in its evolution process Therefore in recent years although large number of forecasting methods have beenproposed and researched in this domain it is very difficult to forecast electricity price with only one traditional model for differentbehaviors of electricity price In this paper we propose an optimized combined forecasting model by ant colony optimizationalgorithm (ACO) based on the generalized autoregressive conditional heteroskedasticity (GARCH) model and support vectormachine (SVM) to improve the forecasting accuracy First both GARCH model and SVM are developed to forecast short-termelectricity price of New South Wales in Australia Then ACO algorithm is applied to determine the weight coefficients Finallythe forecasting errors by three models are analyzed and compared The experiment results demonstrate that the combined modelmakes accuracy higher than the single models

1 Introduction

It is a challenging task and significant role to forecastelectricity price in competitive electricity market Howeverelectricity price has distinct characteristics from other com-modities which is due to its nonlinearity nonstationaritytime variance and uncertain bidding strategy of the marketparticipants All these characteristics can be attributed to thefollowing reasons which distinguish electricity from othercommodities (i) nonstorable nature of electrical energy (ii)the requirement of maintaining constant balance betweendemand and supply (iii) inelastic nature of demand overshort time period and (iv) oligopolistic generation side[1] According to an accurate daily price forecasting in theelectricity market the power suppliers can reduce the cost ofelectricity production optimize the allocation of resourcesreduce the uncertainty of production maximize profits andensure the dominant position in the market competitionAt the same time the consumers can also make a plan to

maximize their utilities and minimize their costs using theelectricity purchased from the pool or using self-productionto protect themselves against high prices

In the current power forecasting researches the forecast-ing of electricity demand and price has emerged as one of themajor research fields in electrical engineering [2] A lot ofresearchers and academicians are engaged in the activity ofdeveloping tools and algorithms for load and price forecast-ing [1] Whereas load forecasting has reached advanced stageof development and load forecasting algorithms with meanabsolute percentage error (MAPE) below 3 are available [34] price-forecasting techniques which are being applied arestill in their early stages of maturity Although a few attemptshave already been made in this direction only qualitativeaspects of price forecasting have been addressed So theimportance and complexity of electricity price forecastingmotivate many researches in this area especially in the recentyears [5]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 504064 9 pageshttpdxdoiorg1011552014504064

2 Abstract and Applied Analysis

The electricity price prediction can be classified into twocategoriesOne is the detailedmarket simulation that requiresplenty of market information The most popular approachis the artificial neural network (ANN) technique ANNtechnique which possesses excellent robustness and errortolerance is an effective way to solve the complex nonlinearmapping problem But the ANN contains a great manyparameters These parameters are always judged by experi-ence so themodel is hard to be established [6] Besides it hasbeen observed thatwhile the neural network (NN) gives smallerror for training patterns the error for testing patterns isusually of larger order [7] in other words when this methodis applied to practical system the accuracy is not good

The other forecasting technology refers to some mathe-matical approaches without a thoroughmarketmodeling butwhich attempt to discover the relation between some knowninputs and the electricity price Arciniegas and ArciniegasRueda [8] applied a Takagi-Sugeno-Kang (TSK) fuzzy infer-ence system to forecast the one-day-ahead real-time peakprice of the Ontario Electricity Market TSKrsquos improvementsin forecasting with respect to ANN and ARMAX are above10 and it has considerable value to forecast one-day-aheadpeak price Fuzzy system does not need to establish a precisemathematical model but its adaptive capacity is limited anda steady-state error may cause oscillations Lei and Feng [9]proposed a novel greymodel to forecast short-term electricityprice for Nord Pool California and Ontario power marketsThe experiment results showed that the forecasting errorhas decreased 1sim6 compared with other grey modelsAlthough grey theory needs very little data it is more suitableto forecast linear data than nonlinear data Wang et al [10]used a combined model based on seasonal adjustment andchaotic theory to forecast electricity price of Austria powermarket A phase space was reconstructed from the time seriesrepresenting the datarsquos chaotic characteristics Particle swarmoptimization (PSO) algorithm was also employed to deter-mine the parameters of chaotic system The forecasting per-formance illustrated that the combined model is better thansingle models The most popular approach is the time seriesalgorithm stationary time series models such as autoregres-sive (AR) [11] dynamic regression and transfer function [12ndash14] and autoregressive integrated moving average (ARIMA)and nonstationary time seriesmodels like generalized autore-gressive conditional heteroskedastic (GARCH) have beenproposed for this purpose Specifically Contreras et al [15]applied ARIMA model to predict the next-day electricityprice of the Spanish electricity market However when facinga particularly big fluctuation time sequence especially het-eroscedastic time series prediction whose variance changesover time ARIMA model does not work well Comparingwith the traditional ARIMA model GARCH model cancommendably predict the condition heteroscedastic timeseries so it is widely applied in the field of finance Garciaet al [16] presented GARCH model to forecast hourly pricesin the electricity markets of Spain and California

Although time series models like ARIMA and GARCHare nonlinear predictors that can meet the condition ofpower price behavior of price signal may not be completelycaptured by the time series techniques To solve this problem

some other artificial methods can be proposed Artificialintelligence (AI) approaches such as neural network andsupport vectormachine (SVM) which have been successfullyapplied in load forecast are also suitable for price forecast[17] Unlike most of the traditional neural network modelswhich implement the empirical risk minimization principleSVM adopts the structural riskminimization principle whichseeks tominimize an upper bound of the generalization errorrather thanminimize the training error [18] SVMpossesses aconcisemathematical form and good generalization ability itcan well solve the practical problems of the small sample sizenonlinearity high dimension and local minimum point

For the forecasting models single model has its ownindependent information of the electric power system andthe proper selection of the individual model can lessen thesystemic information loss Although traditional single timeseries models like AR MA and ARMA are suitable toforecast stable and linear sequences and ARIMA model canbe used to forecast nonstationary and nonlinear time seriesthe forecasting performance is not satisfied In addition thefluctuation of electricity price depends on many complicatedrandom factors in its evolution process and the combinedforecasting model can make full use of the characteristic ofsingle models and reduce the sensitivity of the poorer pre-dictionmethodTherefore an optimized combinedmodel byant colony optimization algorithm (ACO) based on GARCHmodel and SVM model is proposed to predict electricityprice Section 2 introduces the combined forecasting modeltheory GARCH model SVM model and an intelligentoptimization algorithm called ACO In Section 3 a case studyabout forecasting electricity price of New South Wales inAustralia is demonstrated Correspondingly GARCH modeland SVMmodel are utilized to forecast electricity priceThenACO algorithm can determine the weight coefficients Theresults have shown that the optimized combined methodis more reliable than the individual forecasting modelsSection 4 concludes this study

So it is still an essential need to find more accurate androbust approaches for daily electricity price prediction and anoverall assessment of the price-forecasting algorithms is stillrequired

2 Materials and Methods

21 The Optimized Combined Forecasting Model by ACOAlgorithm The combined forecasting theory states that ifthere exist119872 kinds of models to solve a certain forecastingproblem with properly selected weight coefficients severalforecasting methodsrsquo results can be added up Assume that119910

119905(119905 = 1 2 119871) is the actual time series data 119871 is

the number of sample points and |120596119894(119894 = 1 2 119872)

is the weight coefficient for the 119894th forecasting model themathematical model of the combined forecasting model canbe expressed as

119910

119905=

119872

sum

119894=1

120596

119894(119910

119894119905+ 119890

119894119905) 119905 = 1 2 119871

119910

119905=

119872

sum

119894=1

119894119910

119894119905 119905 = 1 2 119871

(1)

Abstract and Applied Analysis 3

where 119894is the estimated value of 120596

119894and 119910

119905is the combined

forecasting valueDetermination of the weight coefficients for each indi-

vidual model is the key step in constructing of a combinedforecasting model This can be achieved by solving anoptimization problem which minimizes the mean absolutepercentage error (MAPE) for the combined model Thisobjective function using can be expressed as

119891 =

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

(2)

st sum119872119894=1120596

119894= 1 0 le 120596

119894le 1 119894 = 1 2 119872

The optimization process can employ ACO algorithm tominimize objective function 119891

22 GARCH Model The ARCH model was initially intro-duced by Engle [19] in order to account for the presence ofheteroscedasticity in economic and financial time series Inan ARCH (q) process the volatility at time 119905 is a functionof the observed data at 119905 minus 1 119905 minus 2 119905 minus 119902 But in thepractical application the ARCH model often needs a verylong condition variance equation and in order to avoidnegative variance parameter estimation it often needs toforcibly demand a fixed hysteresis structure [20] On thisoccasion in order to make ARCH model have long-termmemorizing ability andmore flexible lag structure it is essen-tial to extend ARCHmodel Later Bollerslev [20] introducedthe generalized ARCH (GARCH) process where conditionalvariance not only depends on the squared error term butalso depends on the previous conditional variance [21] AGARCH process of orders 119901 and 119902 denoted as GARCH (pq) (GARCHmodel is used in EVIEWS soft) can be describedas follows [22]

119880

119905| 119865

119905minus1sim 119873 (0 ℎ

119905minus1)

ℎ

119905= 120572

0+ 120572

1119880

2

119905minus1+ sdot sdot sdot + 120572

119902119880

2

119905minus119902+ 120573

1ℎ


119902ℎ

119905minus119902

= 120572

0+

119902

sum

119894=1

120572

119894119880

2

119905minus119894+

119901

sum

119895=1

120573

119895ℎ

119905minus119895

(3)

where 119902 gt 0 1205720gt 0 and 120572

119894ge 0 for 119894 = 1 2 119902 and 120573

119895ge 0

for 119895 = 1 2 119901 Again the conditions 1205720gt 0 120572

119894ge 0 and

120573

119895ge 0 are needed to guarantee that the conditional variance

ℎ

119905ge 0 This study uses GARCH (1 1) model and the (1 1) in

parenthesis indicates that one length of ARCH log(1205721) and

one length of GARCH log(1205731) are used

23 Support Vector Machine (SVM) According to the sta-tistical learning theory and development SVM is based onstructural risk minimization (SRM) principle to minimizethe generalization error the general view is to minimize thetraining error and at the same time minimize the modelcomplexity which has become the cornerstone of modernintelligent algorithm [23]The characteristics of SVMmake ita good candidate model to apply in predicting defect-pronemodules as such conditions are typically encountered

SVM principle is as follows given the training sample(119909

119894 119910

119894) 119909

119894isin 119877

119889 119910

119894isin minus1 +1

119873

119894=1 then the two-class pattern

recognition problem can be cast as the primal problem offinding a hyperplane119908T

119909+119887 = 1 where119908 is a d-dimensionalnormal vector such that these two classes can be separated bytwo margins both parallel to the hyperplane that is for each119909

119894 119894 = 1 2 119873

119908

T119909

119894+ 119887 ge 1 + 120577

119894for 119910119894= +1

119908

T119909

119894+ 119887 le minus1 minus 120577

119894for 119910119894= minus1

(4)

where 120577119894ge 0 119894 = 1 2 119873 are slack variables and 119887 is the

bias This primal problem can easily be cast as the followingquadratic optimization problem [24]

Minimizing 120595 (119908 120577) =

1

2

119908

2+ 119862(

119873

sum

119894=1

120577

119894)

subject to 119910

119894[(119908 sdot 119909

119894) minus 119887] ge 1 minus 120577

119894

(5)

where 120577 = (1205771 120577

2 120577

119873)

The objective of a SVM is to determine the optimalw andoptimal bias b such that the corresponding hyperplane sepa-rates the positive and negative training data with maximummargin and it produces the best generation performanceThishyperplane is called an optimal separating hyperplane [25]

24 Ant Colony Optimization Algorithm (ACO) The antcolony optimization (ACO) developed by Dorigo et al [2627] is a metaheuristic method that aims to find approxi-mate solutions to optimization problems The original ideabehind ant algorithms came from the observations of theforaging behavior of ants and stigmergy Stigmergy is a termthat refers to the indirect communication amongst a self-organizing emergent system by individuals modifying theirlocal environment [28] The detailed steps of ACO algorithm[29] can be described as follows

Step 1 (initializing some parameters) The algorithm starts byinitializing some specific variables such as the maximum ofallowed iterations NCHO and the number of ants ANT andthe initial point which is randomly selected

Step 2 (dividing the definition domain and grouping theants) According to (6) the definition domain is divided intosubregions Each subregion is defined as (7) For all 119894 (119894 =1 2 part) m ants are assigned in every subregion ran-domly Correspondingly each ant is denoted by ant

119894119895(119894 =

1 2 part 119895 = 1 2 119898)

119897

119894=

max119894minusmin

119894

part (6)

grp119894= [min

119894+ 119897

119894sdot (119895 minus 1) min

119894+ 119897

119894sdot 119895] (7)

where 119894 isin [1 119889] 119895 isin [1 part] and d is the dimension of theindependent variables


Step 3 (initializing the pheromone concentration) Each anthas pheromone and the initial pheromone concentration ofjth ant in ith subregion is defined as

120591

119894119895= 119890

(minus119891(119883119894119895))119894 = 1 2 part 119895 = 1 2 119898 (8)

where 119883119894119895represents the location of the jth ant in ith subre-

gion and 119891(119883119894119895) is the objective function If the pheromone is

greater the function value is smaller

Step 4 (determining the elite ant (the optimal ant)) Thepheromone of the elite ant in each group (the local optimalant) can be defined as follows

120591max119894= max (120591

1198941 120591

1198942 120591

119894119895) (9)

where 119894 = 1 2 part and 119895 = 1 2 119898Then based on (10) the pheromone of the global elite ant

(the global optimal ant) can be got

120591

119896119897= max (120591max

1 120591max

2 120591max

119894) (10)

where 119894 = 1 2 part 119896 isin [1 part] and 119897 isin [1119898] Sokth ant in lth subregion is the global elite ant and 119883

119896119897is the

location of the global elite ant

Step 5 (updating the antrsquos location of each group) If |119863119894minus

119863

119895| ge (|119863max minus 119863min|10) then

119863

1015840

119894=

120591

119894119863

119894+ 120572

1

1 minus 120591

119894

120591

119895

(119863

119895minus 119863

119894) if 120572

1lt 05

120591

119894119863

119894+ 120572

1(1 minus 120591

119894)119863

119895 otherwise

(11)

If |119863119894minus 119863

119895| lt (|119863max minus 119863min|10) then let 120583 = |119863

119894minus 119863

119895|

119863

1015840

119894=

119863

119894+ 120583 sdot 120575 sdot 120588 120588 lt 120588

0

119863

119894minus 120583 sdot 120575 sdot 120588 120588 gt 120588

0

(12)

where in this generation119863119895(119895 isin [1119898]) is the location of the

elite ant 120591119895is the pheromone of the elite ant 119863

119894(119894 isin [1119898])

is the location of the common ant and 120591119894is the pheromone of

the common ant 1198631015840119894is the location of the common ant after

updating 1205721isin (minus1 1) 120588 isin (0 1) 120588

0isin (0 1) and 120575 isin 01 times

rand(1)

Step 6 (get the current global optimal solution) Get thecurrent global optimal solution 119891opt(119883119894119895) (119894 = [1 part] 119895 isin[1119898]) based on the new location of the ants and update thepheromone of each ant in each subregion based on

120591

119894119895= 120588120591

119894119895+ 119890

minus119891(119883119894119895) (13)

Step 7 119899119888ℎ119900 = 119899119888ℎ119900 + 1 if 119899119888ℎ119900 gt 119873119862119867119874 go to Step 9otherwise go to Step 8

Step 8 If 119891opt(119883119894119895) (119894 isin [1 part] 119895 isin [1119898]) is less thanthe given solution error condition 10minus2 then go to Step 9otherwise go to Step 4

Step 9 Output 119899119888ℎ119900 and119891opt(119883119894119895) (119894 isin [1 part] 119895 isin [1119898])

3 A Case Study

31 Evaluation Criterion of Forecasting Performance Twoloss functions can be served as the criteria to evaluate theprediction performance relative to electricity price valueincluding mean absolute error (MAE) and mean absolutepercentage error (MAPE) the forecasting effect is betterwhen the loss function value is smallerThe two loss functionsare expressed as follows

MAE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

MAPE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

times 100

(14)

where 119910119905and 119910

119905represent actual and forecasting electricity

price at time and the value of 119871 in our study is 48

32 Simulation and Analysis of Results The proposed opti-mized combined method is tested using a case study aboutforecasting electric price of New South Wales Australia Thedetailed forecasting procedure can be seen in Figure 1 Theelectricity price data were collected on a half-hourly basis (48data points per day starting from 000 AM to 2330 PM) for5 Mondays of electricity price values from February 12 2012to March 11 2012 to predict 1 Monday of electricity price inMarch 18 2012 The actual electricity price data can be seenin Figure 2

GARCH model is operated in EVIEWS soft to forecastelectricity price series of New South Wales Before doingwork this paper judges whether we can apply GARCH (11) model by ARCH Lagrangian Multipliers Test (ARCH LMTest) At the beginning the least square method is used toestimate electricity price data then ARCH test is performedover the residuals by observing the values of the F-statisticswhich is not strong Figure 1 illustrates these procedures Letthe confidence level 120572 be set to 005 If probability P is lessthan 005 the residual sequences will have ARCH effect Inother words it is suitable to use GARCH model Specificallythe results of the ARCH test are presented in Table 1 and itis found that GARCH (1 1) model can be applied to forecastelectricity price

The SVM model which is skilled in dealing with smallsimple and nonlinear data will be used in the next step Thenumber of actual data is 240 (5 Mondays) so the number of119908 is 240 and the detailed values of119908 are seen in Figure 3Thevalue of the bias 119887 = minus02874 After simulation the forecastedvalues can be presented in Table 2

However single traditional model has some limitationswhich cannot present the characteristics of data well There-fore many combined models usually are used to forecastelectricity price in power system In this study an opti-mized combined model based on GARCH and SVM canbe proposed Correspondingly ACO algorithm is presentedto optimize and determine the weight coefficients In ACOalgorithm the experiment uses some parameters as follows119889 = 1 119860119873119879 = 36 119873119862119867119874 = 100 120593 = 05 and 120588 =

05 In fact we are not only interested in simulation of ant


Electricity price data

Establish least square method

Residual test by ARCH test

Establish GARCH model

Output forecasted values

Calculate two class pattern recognition

problems

Find a hyperplane Optimize w andbias b

Outputforecasted values

Build optimized combined model

Calculate globaloptimal pheromone

and find out elite ant

Update the antsrsquolocation of each group

Get the currentglobal optimalUpdate the

pheromone

Iteration

No

Yes

ACO algorithm optimizes weight coefficients

GARCH model

SVM model

Initialize parameterspheromone concentration

and grouping ants

Output nchoand fopt (Xij)

If ncho gt NCHO orfopt (Xij) lt 10minus2

ncho = ncho + 1

solution fopt (Xij)

Figure 1 Flow chart of the optimized combined model

Table 1 ARCH test

ARCH test119865-statistic 1127448 Probability 0000000Obslowast 119877-squared 2290899 Probability 0000000119877-squared 0798223 Mean dependent variance 2575343Adjusted 119877-squared 0797517 SD dependent variance 4527648SE of regression 2037368 Akaike info criterion 8873309Log likelihood minus1271320 Schwarz criterion 8898811Durbin Watson stat 2178122 Probability (119865-statistic) 0000000

0 48 96 144 192 240 2885

10

15

20

25

30

35

40

Elec

tric

ity p

rice (

$AU

)

February February February March March March 12 19 26 4 11 18

Figure 2 The actual electricity price data of six Mondays in NewSouth Wales of Australia in 2012

colonies but also in the use of artificial ant colonies as aone-dimensional optimization tool The objective function is119891 = sum

119871

119905=1(|119910

119905minus 119910

119905|119910

119905) in (2) and 119910

119905= sum

119872

119894=1

119894119910 Its aim

is to optimize 119894so as to minimize the objective function119891

The algorithmwould stop when the number of themaximumiteration is 100 Through simulation and calculation we canget 1= 08058

2= 01942 So the estimated values of the

combined model are written as 119910 = 08058times1199101+01942times119910

2

The concrete values can be seen in Table 2In addition we produce whisker plot with three boxes

which have lines at the lower quartile median and upperquartile values of prediction electricity price value byGARCH (1 1) SVM model and the optimized combinedmodel in Figure 4 It is easy to find that each of the boxesincludes a notch in the position of the median value The

6 Abstract and Applied AnalysisTa

ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

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25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

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0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

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1700

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25027

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2185

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4507

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50885

327

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24901

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31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2

Figure 3 The detailed values of w in the hyperplane

15

20

25

30

35

40

GARCH model SVM model Optimized combinedActual valuesmodel

Figure 4 Box graphs of the actual values and the forecasted valuesby three models

mean values of GARCHmodel and the optimized combinedmodel are closer to the actual values

In order to further illustrate the validity of the combinedmodel two kinds of errors including MAE and MAPE ofGARCH model and SVM model have been listed in Table 2It is clear that MAE andMAPE using GARCH (1 1) model orSVM model are higher than the combined model Althoughthe forecasted values of SVM model are close to the actualvalues before 700 the differences are great large after 700Also it has been observed that the proposed optimized com-binedmodel leads to 018 $AU reductions in total meanMAEand 067 reductions in total mean MAPE respectively incomparison with traditional individual GARCH (1 1) modeland results in 18 $AU reductions in total mean MAE andapproximately 7 reductions in total mean MAPE respec-tively compared to the single SVMmodel Consequently theresults obtained from the optimized combined model agreewith the actual electricity price exceptionally well In otherwords the forecasting model using optimized combinedmodel can yield better results than using GARCH (1 1) andSVMmodel

4 Conclusions

The development of industries agriculture and infrastruc-ture depends on the electric power system electricity con-sumption is relative to modern life consumers want to

minimize their electricity costs and producers expect tomaximize their profits and avoid volatility therefore it isimportant for accurate electricity price prediction

There are four advantages of the optimized combinedmethod At first the optimized combined model by ACOalgorithm based on GARCH (1 1) model and SVM methodfor forecasting half-hourly real-time electricity prices ofNew South Wales creates commendable improvements thatare relatively satisfactorily for current research Second theindividual model has its own independent characteristicin the power system The proper selection of traditionalsingle model can lessen the systemic information loss Theoptimized combined forecasting model can make full useof single models and it is less sensitive to the poorerforecasting approach On the basis there is no doubt that theimproved combined forecasting model performs better thanconventional single model Third an intelligent optimizationalgorithm called ant colony optimization is used to determinethe weight coefficients Finally the optimized combinedmodel is essentially automatic and does not require to makecomplicated decision about the explicit form of models foreach particular case The combined forecasting proceduregives the minimumMAE and MAPE

The final result is that the optimized model has highprediction accuracy and good prediction ability With propercharacteristic selection redundant information is overlookedor even eliminated a more efficient and straightforwardmodel is got To sum up it is clear that the improvedcombinedmodel ismore effective than the existing individualmodels for the electricity price forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Natural Science Founda-tion of P R of China (61300230) the Key Science and Tech-nology Foundation of Gansu Province (1102FKDA010) Nat-ural Science Foundation of Gansu Province (1107RJZA188)and the Fundamental Research Funds for the Central Uni-versities for supporting this research

References

[1] S K Aggarwal L M Saini and A Kumar ldquoElectricity priceforecasting in deregulated markets a review and evaluationrdquoInternational Journal of Electrical Power amp Energy Systems vol31 no 1 pp 13ndash22 2009

[2] DW Bunn ldquoForecasting loads and prices in competitive powermarketsrdquo Proceedings of the IEEE vol 88 no 2 pp 163ndash1692000

[3] R E Abdel-Aal ldquoModeling and forecasting electric daily peakloads using abductive networksrdquo International Journal of Elec-trical Power amp Energy Systems vol 28 no 2 pp 133ndash141 2006

[4] P Mandal T Senjyu N Urasaki and T Funabashi ldquoA neuralnetwork based several-hour-ahead electric load forecasting


using similar days approachrdquo International Journal of ElectricalPower amp Energy Systems vol 28 no 6 pp 367ndash373 2006

[5] N Amjady and M Hemmati ldquoEnergy price forecasting prob-lems and proposals for such predictionsrdquo IEEE Power andEnergy Magazine vol 4 no 2 pp 20ndash29 2006

[6] R Lapedes and R Farber ldquoNonlinear signal processing usingneural networksprediction and system modelingrdquo Tech RepLA-VR87-2662 Los Alamos National Laboratory Los AlamosNM USA 1987

[7] J V E Robert The Application of Neural Networks in theForecasting of Share Prices Finance and Technology Publishing1996

[8] A I Arciniegas and I E Arciniegas Rueda ldquoForecasting short-term power prices in the Ontario Electricity Market (OEM)with a fuzzy logic based inference systemrdquo Utilities Policy vol16 no 1 pp 39ndash48 2008

[9] M L Lei and Z R Feng ldquoA proposed greymodel for short-termelectricity price forecasting in competitive power marketsrdquoInternational Journal of Electrical Power amp Energy Systems vol43 no 1 pp 531ndash538 2012

[10] J Wang H Lu Y Dong and D Chi ldquoThe model of chaoticsequences based on adaptive particle swarm optimization arith-metic combined with seasonal termrdquo Applied MathematicalModelling vol 36 no 3 pp 1184ndash1196 2012

[11] O B Fosso A Gjelsvik A Haugstad B Mo and I Wan-gensteen ldquoGeneration scheduling in a deregulated system thenorwegian caserdquo IEEE Transactions on Power Systems vol 14no 1 pp 75ndash80 1999

[12] H Zareipour C A Canzares K Bhattacharya and JThomsonldquoApplication of public-domain market information to forecastOntariorsquos wholesale electricity pricesrdquo IEEE Transactions onPower Systems vol 21 no 4 pp 1707ndash1717 2006

[13] F J Nogales J Contreras A J Conejo and R EspınolaldquoForecasting next-day electricity prices by time series modelsrdquoIEEE Transactions on Power Systems vol 17 no 2 pp 342ndash3482002

[14] F J Nogales and A J Conejo ldquoElectricity price forecastingthrough transfer function modelsrdquo Journal of the OperationalResearch Society vol 57 no 4 pp 350ndash356 2006

[15] J Contreras R Espınola F J Nogales and A J ConejoldquoARIMA models to predict next-day electricity pricesrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1014ndash10202003

[16] R C Garcia J Contreras M van Akkeren and J B C GarcialdquoA GARCH forecasting model to predict day-ahead electricitypricesrdquo IEEE Transactions on Power Systems vol 20 no 2 pp867ndash874 2005

[17] B R Szkuta L A Sanabria and T S Dillon ldquoElectricity priceshort-term forecasting using artificial neural networksrdquo IEEETransactions on Power Systems vol 14 no 3 pp 851ndash857 1999

[18] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[19] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[20] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[21] C-H Tseng S-T Cheng and Y-H Wang ldquoNew hybridmethodology for stock volatility predictionrdquo Expert Systemswith Applications vol 36 no 2 pp 1833ndash1839 2009


[23] D Niu D Liu and D D Wu ldquoA soft computing system forday-ahead electricity price forecastingrdquoApplied Soft ComputingJournal vol 10 no 3 pp 868ndash875 2010

[24] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995

[25] H-J Lin and J P Yeh ldquoOptimal reduction of solutions for sup-port vector machinesrdquo Applied Mathematics and Computationvol 214 no 2 pp 329ndash335 2009

[26] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 26 no 1 pp29ndash41 1996

[27] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[28] M J Meena K R Chandran A Karthik and A V Samuel ldquoAnenhanced ACO algorithm to select features for text categoriza-tion and its parallelizationrdquo Expert Systems with Applicationsvol 39 no 5 pp 5861ndash5871 2012

[29] Q Y Li Q B Zhu and W Ma ldquoGrouped ant colony algorithmfor solving continuous optimization problemsrdquoComputer Engi-neering and Applications vol 46 no 30 pp 46ndash49 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The electricity price prediction can be classified into twocategoriesOne is the detailedmarket simulation that requiresplenty of market information The most popular approachis the artificial neural network (ANN) technique ANNtechnique which possesses excellent robustness and errortolerance is an effective way to solve the complex nonlinearmapping problem But the ANN contains a great manyparameters These parameters are always judged by experi-ence so themodel is hard to be established [6] Besides it hasbeen observed thatwhile the neural network (NN) gives smallerror for training patterns the error for testing patterns isusually of larger order [7] in other words when this methodis applied to practical system the accuracy is not good

The other forecasting technology refers to some mathe-matical approaches without a thoroughmarketmodeling butwhich attempt to discover the relation between some knowninputs and the electricity price Arciniegas and ArciniegasRueda [8] applied a Takagi-Sugeno-Kang (TSK) fuzzy infer-ence system to forecast the one-day-ahead real-time peakprice of the Ontario Electricity Market TSKrsquos improvementsin forecasting with respect to ANN and ARMAX are above10 and it has considerable value to forecast one-day-aheadpeak price Fuzzy system does not need to establish a precisemathematical model but its adaptive capacity is limited anda steady-state error may cause oscillations Lei and Feng [9]proposed a novel greymodel to forecast short-term electricityprice for Nord Pool California and Ontario power marketsThe experiment results showed that the forecasting errorhas decreased 1sim6 compared with other grey modelsAlthough grey theory needs very little data it is more suitableto forecast linear data than nonlinear data Wang et al [10]used a combined model based on seasonal adjustment andchaotic theory to forecast electricity price of Austria powermarket A phase space was reconstructed from the time seriesrepresenting the datarsquos chaotic characteristics Particle swarmoptimization (PSO) algorithm was also employed to deter-mine the parameters of chaotic system The forecasting per-formance illustrated that the combined model is better thansingle models The most popular approach is the time seriesalgorithm stationary time series models such as autoregres-sive (AR) [11] dynamic regression and transfer function [12ndash14] and autoregressive integrated moving average (ARIMA)and nonstationary time seriesmodels like generalized autore-gressive conditional heteroskedastic (GARCH) have beenproposed for this purpose Specifically Contreras et al [15]applied ARIMA model to predict the next-day electricityprice of the Spanish electricity market However when facinga particularly big fluctuation time sequence especially het-eroscedastic time series prediction whose variance changesover time ARIMA model does not work well Comparingwith the traditional ARIMA model GARCH model cancommendably predict the condition heteroscedastic timeseries so it is widely applied in the field of finance Garciaet al [16] presented GARCH model to forecast hourly pricesin the electricity markets of Spain and California

Although time series models like ARIMA and GARCHare nonlinear predictors that can meet the condition ofpower price behavior of price signal may not be completelycaptured by the time series techniques To solve this problem

some other artificial methods can be proposed Artificialintelligence (AI) approaches such as neural network andsupport vectormachine (SVM) which have been successfullyapplied in load forecast are also suitable for price forecast[17] Unlike most of the traditional neural network modelswhich implement the empirical risk minimization principleSVM adopts the structural riskminimization principle whichseeks tominimize an upper bound of the generalization errorrather thanminimize the training error [18] SVMpossesses aconcisemathematical form and good generalization ability itcan well solve the practical problems of the small sample sizenonlinearity high dimension and local minimum point

For the forecasting models single model has its ownindependent information of the electric power system andthe proper selection of the individual model can lessen thesystemic information loss Although traditional single timeseries models like AR MA and ARMA are suitable toforecast stable and linear sequences and ARIMA model canbe used to forecast nonstationary and nonlinear time seriesthe forecasting performance is not satisfied In addition thefluctuation of electricity price depends on many complicatedrandom factors in its evolution process and the combinedforecasting model can make full use of the characteristic ofsingle models and reduce the sensitivity of the poorer pre-dictionmethodTherefore an optimized combinedmodel byant colony optimization algorithm (ACO) based on GARCHmodel and SVM model is proposed to predict electricityprice Section 2 introduces the combined forecasting modeltheory GARCH model SVM model and an intelligentoptimization algorithm called ACO In Section 3 a case studyabout forecasting electricity price of New South Wales inAustralia is demonstrated Correspondingly GARCH modeland SVMmodel are utilized to forecast electricity priceThenACO algorithm can determine the weight coefficients Theresults have shown that the optimized combined methodis more reliable than the individual forecasting modelsSection 4 concludes this study

So it is still an essential need to find more accurate androbust approaches for daily electricity price prediction and anoverall assessment of the price-forecasting algorithms is stillrequired

2 Materials and Methods

21 The Optimized Combined Forecasting Model by ACOAlgorithm The combined forecasting theory states that ifthere exist119872 kinds of models to solve a certain forecastingproblem with properly selected weight coefficients severalforecasting methodsrsquo results can be added up Assume that119910

119905(119905 = 1 2 119871) is the actual time series data 119871 is

the number of sample points and |120596119894(119894 = 1 2 119872)

is the weight coefficient for the 119894th forecasting model themathematical model of the combined forecasting model canbe expressed as

119910

119905=

119872

sum

119894=1

120596

119894(119910

119894119905+ 119890

119894119905) 119905 = 1 2 119871

119910

119905=

119872

sum

119894=1

119894119910

119894119905 119905 = 1 2 119871

(1)



119894and 119910




119891 =

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

(2)

st sum119872119894=1120596

119894= 1 0 le 120596

119894le 1 119894 = 1 2 119872



119880

119905| 119865

119905minus1sim 119873 (0 ℎ

119905minus1)

ℎ

119905= 120572

0+ 120572

1119880

2


119902119880

2

119905minus119902+ 120573

1ℎ


119902ℎ

119905minus119902

= 120572

0+

119902

sum

119894=1

120572

119894119880

2

119905minus119894+

119901

sum

119895=1

120573

119895ℎ

119905minus119895

(3)

where 119902 gt 0 1205720gt 0 and 120572

119894ge 0 for 119894 = 1 2 119902 and 120573

119895ge 0


119894ge 0 and

120573


ℎ






119894 119910

119894) 119909

119894isin 119877

119889 119910


119873




119894 119894 = 1 2 119873

119908

T119909

119894+ 119887 ge 1 + 120577

119894for 119910119894= +1

119908

T119909

119894+ 119887 le minus1 minus 120577

119894for 119910119894= minus1

(4)



Minimizing 120595 (119908 120577) =

1

2

119908

2+ 119862(

119873

sum

119894=1

120577

119894)

subject to 119910

119894[(119908 sdot 119909

119894) minus 119887] ge 1 minus 120577

119894

(5)

where 120577 = (1205771 120577

2 120577

119873)





119894119895(119894 =

1 2 part 119895 = 1 2 119898)

119897

119894=

max119894minusmin

119894

part (6)

grp119894= [min

119894+ 119897

119894sdot (119895 minus 1) min

119894+ 119897

119894sdot 119895] (7)




120591

119894119895= 119890

(minus119891(119883119894119895))119894 = 1 2 part 119895 = 1 2 119898 (8)





120591max119894= max (120591

1198941 120591

1198942 120591

119894119895) (9)



120591

119896119897= max (120591max

1 120591max

2 120591max

119894) (10)


119896119897is the



119863


119863

1015840

119894=

120591

119894119863

119894+ 120572

1

1 minus 120591

119894

120591

119895

(119863

119895minus 119863

119894) if 120572

1lt 05

120591

119894119863

119894+ 120572

1(1 minus 120591

119894)119863

119895 otherwise

(11)

If |119863119894minus 119863


119894minus 119863

119895|

119863

1015840

119894=

119863

119894+ 120583 sdot 120575 sdot 120588 120588 lt 120588

0

119863


0

(12)



119894(119894 isin [1119898])





rand(1)


120591

119894119895= 120588120591

119894119895+ 119890

minus119891(119883119894119895) (13)




3 A Case Study


MAE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

MAPE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

times 100

(14)

where 119910119905and 119910















problems








pheromone

Iteration

No

Yes


GARCH model

SVM model


and grouping ants



ncho = ncho + 1

solution fopt (Xij)


Table 1 ARCH test


0 48 96 144 192 240 2885

10

15

20

25

30

35

40

Elec

tric

ity p

rice (

$AU

)




119871

119905=1(|119910

119905minus 119910

119905|119910

119905) in (2) and 119910

119905= sum

119872

119894=1

119894119910 Its aim





2




ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

2437

25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

25031

0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

1538

25968

0118

046

1700

2648

25027

1453

549

32604

6124

2313

26498

0018

007

1730

2763

24987

2643

957

30468

2838

1027

26052

1578

571

1800

2707

24885

2185

807

31577

4507

1665

2618

50885

327

1830

2717

24901

2269

835

31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and 119910




119891 =

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

(2)

st sum119872119894=1120596

119894= 1 0 le 120596

119894le 1 119894 = 1 2 119872



119880

119905| 119865

119905minus1sim 119873 (0 ℎ

119905minus1)

ℎ

119905= 120572

0+ 120572

1119880

2


119902119880

2

119905minus119902+ 120573

1ℎ


119902ℎ

119905minus119902

= 120572

0+

119902

sum

119894=1

120572

119894119880

2

119905minus119894+

119901

sum

119895=1

120573

119895ℎ

119905minus119895

(3)

where 119902 gt 0 1205720gt 0 and 120572

119894ge 0 for 119894 = 1 2 119902 and 120573

119895ge 0


119894ge 0 and

120573


ℎ






119894 119910

119894) 119909

119894isin 119877

119889 119910


119873




119894 119894 = 1 2 119873

119908

T119909

119894+ 119887 ge 1 + 120577

119894for 119910119894= +1

119908

T119909

119894+ 119887 le minus1 minus 120577

119894for 119910119894= minus1

(4)



Minimizing 120595 (119908 120577) =

1

2

119908

2+ 119862(

119873

sum

119894=1

120577

119894)

subject to 119910

119894[(119908 sdot 119909

119894) minus 119887] ge 1 minus 120577

119894

(5)

where 120577 = (1205771 120577

2 120577

119873)





119894119895(119894 =

1 2 part 119895 = 1 2 119898)

119897

119894=

max119894minusmin

119894

part (6)

grp119894= [min

119894+ 119897

119894sdot (119895 minus 1) min

119894+ 119897

119894sdot 119895] (7)




120591

119894119895= 119890

(minus119891(119883119894119895))119894 = 1 2 part 119895 = 1 2 119898 (8)





120591max119894= max (120591

1198941 120591

1198942 120591

119894119895) (9)



120591

119896119897= max (120591max

1 120591max

2 120591max

119894) (10)


119896119897is the



119863


119863

1015840

119894=

120591

119894119863

119894+ 120572

1

1 minus 120591

119894

120591

119895

(119863

119895minus 119863

119894) if 120572

1lt 05

120591

119894119863

119894+ 120572

1(1 minus 120591

119894)119863

119895 otherwise

(11)

If |119863119894minus 119863


119894minus 119863

119895|

119863

1015840

119894=

119863

119894+ 120583 sdot 120575 sdot 120588 120588 lt 120588

0

119863


0

(12)



119894(119894 isin [1119898])





rand(1)


120591

119894119895= 120588120591

119894119895+ 119890

minus119891(119883119894119895) (13)




3 A Case Study


MAE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

MAPE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

times 100

(14)

where 119910119905and 119910















problems








pheromone

Iteration

No

Yes


GARCH model

SVM model


and grouping ants



ncho = ncho + 1

solution fopt (Xij)


Table 1 ARCH test


0 48 96 144 192 240 2885

10

15

20

25

30

35

40

Elec

tric

ity p

rice (

$AU

)




119871

119905=1(|119910

119905minus 119910

119905|119910

119905) in (2) and 119910

119905= sum

119872

119894=1

119894119910 Its aim





2




ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

2437

25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

25031

0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

1538

25968

0118

046

1700

2648

25027

1453

549

32604

6124

2313

26498

0018

007

1730

2763

24987

2643

957

30468

2838

1027

26052

1578

571

1800

2707

24885

2185

807

31577

4507

1665

2618

50885

327

1830

2717

24901

2269

835

31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591

119894119895= 119890

(minus119891(119883119894119895))119894 = 1 2 part 119895 = 1 2 119898 (8)





120591max119894= max (120591

1198941 120591

1198942 120591

119894119895) (9)



120591

119896119897= max (120591max

1 120591max

2 120591max

119894) (10)


119896119897is the



119863


119863

1015840

119894=

120591

119894119863

119894+ 120572

1

1 minus 120591

119894

120591

119895

(119863

119895minus 119863

119894) if 120572

1lt 05

120591

119894119863

119894+ 120572

1(1 minus 120591

119894)119863

119895 otherwise

(11)

If |119863119894minus 119863


119894minus 119863

119895|

119863

1015840

119894=

119863

119894+ 120583 sdot 120575 sdot 120588 120588 lt 120588

0

119863


0

(12)



119894(119894 isin [1119898])





rand(1)


120591

119894119895= 120588120591

119894119895+ 119890

minus119891(119883119894119895) (13)




3 A Case Study


MAE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

MAPE = 1

119871

119871

sum

119905=1

1003816

1003816

1003816

1003816

119910

119905minus 119910

119905

1003816

1003816

1003816

1003816

119910

119905

times 100

(14)

where 119910119905and 119910















problems








pheromone

Iteration

No

Yes


GARCH model

SVM model


and grouping ants



ncho = ncho + 1

solution fopt (Xij)


Table 1 ARCH test


0 48 96 144 192 240 2885

10

15

20

25

30

35

40

Elec

tric

ity p

rice (

$AU

)




119871

119905=1(|119910

119905minus 119910

119905|119910

119905) in (2) and 119910

119905= sum

119872

119894=1

119894119910 Its aim





2




ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

2437

25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

25031

0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

1538

25968

0118

046

1700

2648

25027

1453

549

32604

6124

2313

26498

0018

007

1730

2763

24987

2643

957

30468

2838

1027

26052

1578

571

1800

2707

24885

2185

807

31577

4507

1665

2618

50885

327

1830

2717

24901

2269

835

31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















problems








pheromone

Iteration

No

Yes


GARCH model

SVM model


and grouping ants



ncho = ncho + 1

solution fopt (Xij)


Table 1 ARCH test


0 48 96 144 192 240 2885

10

15

20

25

30

35

40

Elec

tric

ity p

rice (

$AU

)




119871

119905=1(|119910

119905minus 119910

119905|119910

119905) in (2) and 119910

119905= sum

119872

119894=1

119894119910 Its aim





2




ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

2437

25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

25031

0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

1538

25968

0118

046

1700

2648

25027

1453

549

32604

6124

2313

26498

0018

007

1730

2763

24987

2643

957

30468

2838

1027

26052

1578

571

1800

2707

24885

2185

807

31577

4507

1665

2618

50885

327

1830

2717

24901

2269

835

31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ble2Fo

recaste

dresults

comparis

onsa

mon

gthreem

odels

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

000

261

24991

1109

425

2710

510

05385

25402

0698

268

030

2253

25000

2470

1096

26405

3875

1720

25273

2743

1218

100

2123

24902

3672

1730

2637

45144

2423

2518

83958

1864

130

2133

24779

344

81616

25413

4083

1914

24901

3571

1674

200

1612

24772

8652

5367

24263

8143

5052

24673

8553

5306

230

1979

24268

4478

2263

22438

264

81338

23912

4122

2083

300

1918

24540

5360

2794

17846

1334

695

23240

406

02117

330

1991

24547

4637

2329

20014

0104

052

23667

3757

1887

400

2047

24612

4142

2023

17685

2785

1360

23267

2797

1366

430

1991

24671

4761

2391

18550

1360

683

23482

3572

1794

500

1991

24631

4721

2371

22415

2505

1258

24201

4291

2155

530

2008

24626

4546

2264

26576

6496

3235

25005

4925

2453

600

2496

24640

0320

0128

20701

4259

1706

23875

1085

435

630

1947

24898

5428

2788

2176

52295

1179

24290

4820

2475

700

2557

24608

0962

376

24454

1116

436

24578

0992

388

730

2716

24871

2289

843

24403

2757

1015

24780

2380

876

800

2623

24890

1340

511

2617

80052

020

2514

010

90416

830

2732

24959

2361

864

26494

0826

302

25257

2063

755

900

314

24904

6496

2069

27014

4386

1397

25314

6086

1938

930

2875

24502

4248

1478

2815

20598

208

25211

3539

1231

1000

2876

24631

4129

1436

27438

1323

460

25176

3584

1246

1030

2699

24677

2313

857

29447

2457

910

25604

1386

514

1100

2652

24831

1689

637

29883

3363

1268

25812

0708

267

1130

2594

24920

1020

393

27960

2020

779

25510

0430

166

1200

2685

24985

1865

695

2953

32683

999

25868

0982

366

1230

2621

24949

1261

481

3131

55105

1948

2618

50025

010

1300

2529

24981

0309

122

3212

06830

2701

26367

1077

426

1330

2437

25028

0658

270

30774

640

42628

2614

417

7472

81400

2445

25031

0581

238

32635

8185

3348

26508

2058

842

1430

2203

25034

300

41364

33094

11064

5022

26599

4569

2074

1500

2509

24860

0230

092

3618

011090

4420

27058

1968

784

1530

2434

24988

064

8266

31713

7373

3029

26294

1954

803

1600

2533

25021

0309

122

32692

7362

2906

26510

1180

466

1630

2585

25039

0811

314

29826

3976

1538

25968

0118

046

1700

2648

25027

1453

549

32604

6124

2313

26498

0018

007

1730

2763

24987

2643

957

30468

2838

1027

26052

1578

571

1800

2707

24885

2185

807

31577

4507

1665

2618

50885

327

1830

2717

24901

2269

835

31612

444

21635

26205

0965

355

1900

273

24899

2401

880

32823

5523

2023

26438

0862

316

1930

2623

24888

1342

512

32078

5848

2230

26284

0054

021

2000

2537

24959

0412

162

31456

6086

2399

26220

0850

335

2030

2304

25019

1979

859

3032

072

803160

26048

3008

1306

2100

201

24946

4846

2411

29348

9248

4601

25801

5701

2836


Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Con

tinued

Times

Actualvalues

GARC

Hmod

elSV

Mmod

elCom

binedmod

elFo

recaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

Forecaste

dvalues

MAE($AU

)MAPE

()

2130

2417

24672

0502

208

27873

3703

1532

25293

1123

465

2200

2098

24909

3929

873

25414

4434

2114

25007

4027

1920

2230

2509

24755

0335

134

25668

0578

231

24932

0158

063

2300

234

24946

1546

661

24871

1471

629

24931

1531

654

2330

2375

24962

1212

510

25736

1986

836

25113

1363

574

Meanerror

253

1133

415

1765

235

1066


0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 48 96 144 192 240

0

2

4

6

8

minus6

minus4

minus2


15

20

25

30

35

40





4 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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