research article midterm electricity market clearing price … · 2018. 11. 30. · research...

Research ArticleMidterm Electricity Market Clearing Price Forecasting UsingTwo-Stage Multiple Support Vector Machine

Xing Yan and Nurul A Chowdhury

Department of Electrical Engineering University of Saskatchewan Saskatoon SK Canada S7N 5A9

Correspondence should be addressed to Xing Yan xiy201mailusaskca

Received 31 May 2014 Revised 14 January 2015 Accepted 15 January 2015

Academic Editor Chuanwen Jiang

Copyright copy 2015 X Yan and N A ChowdhuryThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Currently there are many techniques available for short-term forecasting of the electricity market clearing price (MCP) but verylittle work has been done in the area of midterm forecasting of the electricity MCP The midterm forecasting of the electricityMCP is essential for maintenance scheduling planning bilateral contracting resources reallocation and budgeting A two-stagemultiple support vectormachine (SVM) basedmidterm forecastingmodel of the electricityMCP is proposed in this paperThe firststage is utilized to separate the input data into corresponding price zones by using a single SVMThen the second stage is appliedutilizing four parallel designed SVMs to forecast the electricity price in four different price zones Compared to the forecastingmodel using a single SVM the proposed model showed improved forecasting accuracy in both peak prices and overall systemPJM interconnection data are used to test the proposed model

1 Introduction

Since the generating companies as well as the bulk sellerswant to maximize their profits under a deregulated electricitymarket offering the appropriate amount of electricity at theright timewith the right bidding price is of paramount impor-tance The forecasting of the electricity price is a predictionof future electricity price based on given forecast electricitydemand temperature sunshine fuel cost precipitation andother related factors The electricity market clearing price(MCP) also called the equilibrium price exists when anelectricity market is clear of shortage and surplus Once theelectricity MCP is determined every supplier whose offeringprice is below or equal to the electricity MCP will be pickedup to supply electricity at that hour To ensure fairness ofthe market and to avoid market manipulation the picked upsuppliers will be paid at the electricityMCP not the price theyoffered

Currently in the electricity price forecasting studiesshort-term forecasting of the electricity MCP is the mostfocused research area It is also commonly known as the 24-hour day-ahead electricity price forecastingUnlike the short-term forecasting very little work has been done to forecastelectricity MCP on a midterm basis [1ndash5] The midterm

forecasting of the electricity MCP focuses on a time framefrom one month to six months It is essential for decisionmaking and midterm planning purposes such as generationplant expansion and maintenance schedule reallocation ofresources bilateral contracts and hedging strategies [5] Themidterm forecasting of the electricity MCP is different fromthe short-term forecasting inmanyways First of all midtermforecasting by its nature cannot utilize the trend from theimmediate past while the short-term forecasting can Futuresegment is not contiguous to the immediate past history forwhich electricity MCP data are available for the midtermforecasting of the electricity MCP In order to accuratelyforecast the future electricity MCP with segmented inputdata themidterm forecastingmodelmust possess very strongadaptability in handling out-of-sample and segmented dataduring training phase Secondly because of the unavailabilityof immediate past data forecasting technique such as timeseries cannot be utilized in midterm forecasting In midtermelectricity price forecasting each input data at hour 119905 hasto be treated independently from its nearby data such as119905 minus 119899 or 119905 + 119899 for 119899 equals 1 2 3 Moreover locatingand predicating peak prices in the midterm forecasting ofthe electricity MCP are extremely difficult As the short-term forecasting can utilize the trend from the immediate

Hindawi Publishing CorporationJournal of EnergyVolume 2015 Article ID 384528 11 pageshttpdxdoiorg1011552015384528

2 Journal of Energy

past locating the peak prices in most cases is usually veryaccurateThemain challenge in short-term forecasting is howaccurately the forecasting model can predict the values ateach peak price spot However in the midterm forecastingof the electricity MCP because of the unavailability of datafrom the immediate past locating the peak prices becomesextremely difficult As a result accurately predicting thevalues at peak price spots becomes even more difficultThe fourth difference is the length of historical data thatare needed to train the forecasting model The short-termforecasting of the electricity price model usually only needsthe data of the last couple of days to train the forecastingmodel On the other hand the midterm forecasting modelrequires one year of historical data in order to train theforecasting model Finally only nonlinear regression basedforecastingmodels are capable ofmidterm forecasting Linearregression based forecasting models can only be consideredas an add-on module in the midterm forecasting

Before hybrid models utilized in the forecasting of theelectricity MCP autoregressive integrated moving average(ARIMA) [6] wavelet transform [7ndash9] Monte Carlo sim-ulation [10] bid-based stochastic model [10] time series[11 12] and dynamic regression [13] are the first generationof techniques utilized in the forecasting of the electricityMCP Artificial neural network (ANN) was later on appliedto forecast short-term electricity MCP [2ndash4 14ndash19] becauseof its flexibility in handling highly nonlinear relationshipsand relatively easy implementation PJM interconnectionAustralian electricity market England-Wales pool and NewEngland ISO [16] are currently utilizing ANN to forecast theelectricity MCP

Support vector machine (SVM) is a new learningmethodbased on structural risk minimization and has gainedincreased attention in recent electricity price forecasting [20ndash23] studies The major advantages of SVM over ANN or anyother forecasting models are that SVM can avoid problemssuch as data overfitting local minimum and unpredictablylarge out-of-sample data error while at the same time stillachieving better results SVM is also a very robust forecastingmodel and will always end up with the same acceptableresult regardless of the initial values On top of that SVMhas less adjustable parameters compared to ANN and isless complicated in control parameter selection In electricityprice forecasting a traditional SVM can achieve around 3[24] improved forecasting accuracy compared to a traditionalANN Several algorithms including genetic algorithm (GA)[24ndash27] artificial fish swarm algorithm [28] independentcomponent analysis (ICA) algorithm [29 30] and rough setsalgorithm [31 32] are used to further optimize the trainingof SVM Moreover the least squares support vector machine(LSSVM) was later developed to improve the accuracy of theoriginal SVM [25 27 33ndash35] Although each method hasshown some improvements the overall system accuracy wasstill quite low

The new trend in recent short-term forecasting of theelectricity price studies is using hybrid models combiningseveral prediction methods Hybrid models can compensatetheweaknesses of utilizing any individual establishedmethodand achieve better overall system results Torghaban et al [5]

proposed hybrid midterm electricity monthly average priceforecasting models combined with SVMSVM SVMNNNNSVM and NNNN Swief et al [23] utilized the principlecomponent analysis (PCA) and 119896 nearest neighbor (119896NN)points technique to reduce the number of data entries to theSVM model Wang and Qin [32] introduced an electricityprice prediction model using rough sets and SVM wherethe rough sets theory was used to simplify the SVM inputvariables Xie et al [33] proposed a hybrid electricity priceforecasting model that integrates clustering algorithm withLSSVM Fan et al [35] proposed a hybrid forecasting modelthat includes the Bayesian clustering by dynamics (BCD)and SVM where the BCD classifier is applied to clusterthe input data set into several subsets in an unsupervisedmanner Zhao et al [36] forecasted the prediction intervalof the electricity price utilizing SVM and nonlinear con-ditional heteroscedastic forecasting (NCHF) model Theyalso proposed a hybrid model [37 38] using SVM andprobability classifier to predict the price spike occurrenceand values Areekul et al [39] proposed forecast techniquesbased on autoregressive integratedmoving average (ARIMA)and ANN Xie et al [40] proposed a sensitivity analysis ofelectricity price utilizing SVM and regression model whereseveral price-load elasticity equations are built based on theprice pattern data sets classified by SVM classification Asimilar work focused on price-load elasticity analysis is doneby Wang in [41] Pousinho et al [42] proposed a hybridshort-term electricity price forecastingmodel combined withparticle swarm optimization (PSO) and adaptive-networkbased fuzzy inference system (ANFIS)

A new midterm forecasting model of the electricityMCP utilizing two-stage multiple SVM is proposed in thispaper based on the studying of previous works in midtermforecasting of the electricity MCP shown in Table 1 Theproposed model can predict both patterns and values ofelectricity MCP A single SVM is first used to separate theinput data into corresponding price zonesThen four paralleldesigned SVMs are used to forecast the price values insideeach separated price zone The main contributions of thispaper include (1) addressing and resolving problems asso-ciated with midterm forecasting of the electricity MCP (2)addressing and resolving the problems associated with usinga single nonlinear forecasting model to forecast midtermelectricity MCP such as SVM and (3) presenting a two-stage multiple SVM based forecasting model Experimentalresults using historical data from the PJM interconnectionsystem demonstrated that the proposed two-stage multipleSVM forecasting model can improve the price predictingaccuracy in both low and peak price zones The rest ofthe paper is organized as follows Two-stage multiple SVMbased midterm forecasting model of the electricity MCP isgiven in Section 2 Case studies are presented in Section 3Conclusions are included in Section 4

2 Two-Stage Multiple SVM Based MidtermForecasting Model of the Electricity MCP

21 Support Vector Machine SVM was first introduced byVladimir Vapnik in 1979 based on the statistical learning and

Journal of Energy 3

Table 1 Comparison of previously published midterm forecasting techniques of electricity prices

Authors Forecasting objectives and techniques Strengths Weaknesses

Torbaghan et al [1]Midterm forecasting of the monthlyaverage electricity spot prices usingvarious techniques

Both single and hybrid modelsare tested

Only focused on the monthlyaverage electricity spot prices(only 12 outputs)

Yan [2] Yan andChowdhury [3 4]

Midterm forecasting of the electricityhourly MCP utilizing ANN technique Extended training data set Poor performance in forecasting

peak prices

Torghaban et al [5]Midterm forecasting of the monthlyaverage electricity spot prices utilizinglinear forecasting models

Dummy variables are consideredin the model indicatingseasonality during simulation


Yan andChowdhury [43]

Midterm forecasting of the electricityhourly MCP utilizing hybrid LSSVMand ARMAX techniques

The proposed hybrid model iscapable of adjusting the errorsfrom the previous module

Limited improvements inforecasting peak prices

Pedregal andTrapero [44]

Midterm forecasting of the electricityhourly price utilizing unobservedcomponent models

Utilization of numbers framedshort-term forecast results toserve in the midterm forecasting

The proposed model is comparedwith an ARIMA model whichhas quite weak performance inmidterm forecasting

Gonzalez et al [45]

Midterm forecasting of electricity hourprice utilizing hybrid approachesbased on the analysis between marketprice and marginal costs

Forecast the real price based onadjusting the prediction values ofthe marginal price forecasting

Only price data is involved in theproposed study and the forecastof the marginal price is under anideal electricity market

X = [x1 x2 xN]

x1

x2

xN

N inputs

sum

Bias

Output neuron

Hidden layer

K(x xi)

K(x xi)

K(x xi)

Figure 1 SVM architecture [36]

later on developed by Vladimir Vapnik and his coworkersat the ATampT Bell Laboratories in 1995 After the book AnIntroduction of Support Vector Machines and Other Kernel-Based LearningMethods byNelloCristianini and John Shawe-Taylor published in 2000 it started getting more popularityand application in many fields At the early stage SVM wasonly used for classification purposes Then the regressioncomputation of nonlinear function was added by solving aconvex quadratic optimization problem [46]

A brief SVM architecture is shown in Figure 1 It isvery similar to a 3-layer feed-forward ANN It has theinput and output layers containing either single or multipleinputoutput data The only difference is inside the hiddenlayer where kernels replace the hidden neuronsThe workingprinciple of a SVM is also different from that of an ANNA 3-layer feed-forward ANN contains two transfer functionsconnecting the input layer to the hidden layer and the hiddenlayer to the output layer SVM on the other hand onlyhas kernels acting like transfer functions inside the hidden

layer connecting the input layer and the output layer Kernelstransfer lowdimensional input data vector into amuchhigherdimensional vector (sometime can be infinite) and eventuallytransfer the highly nonlinear problem inside the input spaceinto a linear problem inside the feature space After thetransformation is completed optimization algorithms arethen applied in order to perform the regression or classifi-cation computation SVM uses a quadratic formulation asoptimization algorithm

Suppose that (119883119905 119910119905) for 119905 = 1 to 119873 is a given set of

data where119883119905= (1199091199051 1199091199052 119909

119905119896) is the input vector at time

119905 with 119896 elements and 119910119905is the corresponding price data at

time 119905 which could be defined as

119910119905= 119891 (119883

119905) = ⟨119882 120593 (119883

119905)⟩ + 119887 (1)

where ⟨ ⟩ denotes the dot product119882 is the weight vector 119887is the bias and 120593() is the mapping function transfers inputvector 119883

119905into a much higher dimensional feature space

(could be infinite) The corresponding optimization problemis then

minimize 1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

subject to 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887 le 120576 + 120585

119905

⟨119882 120593 (119883119905)⟩ + 119887 minus 119910

119905le 120576 + 120585

lowast

119905

120585119905 120585lowast

119905ge 0

(2)

where 119862 is the regularization constant with regard to the unitcost of errors 120585

119905and 120585lowast119905are the slack variables measuring the

cost of the errors both above and below the target value in

4 Journal of Energy

the training points The 120576-insensitive loss function with 2120576bandwidth is defined as

1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816120576=

0 if 1003816100381610038161003816119910119905 minus 119891 (119883119905)1003816100381610038161003816le 120576

1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816minus 120576 otherwise

(3)

When adding the Lagrange multipliers to (2) the problemcan be rewritten as

119871 (119882 120585 120585lowast 120572 120572lowast 120578 120578lowast)

=1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

minus

119873

sum

119905=1

120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887)

minus

119873

sum

119905=1

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887)

minus

119873

sum

119905=1

(120578119905120585119905+ 120578lowast

119905120585lowast

119905)

(4)

where 120572119905 120572lowast119905 120578119905 and 120578lowast

119905are the Lagrange multipliers The

partial derivates of the Lagrange function with the primalvariables (119882 119887 120585

119905 and 120585lowast

119905) are then

120597119871SVM120597119882

= 0 997888rarr 119882 =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) 120593 (119883

119905)

120597119871SVM120597119887

= 0 997888rarr

119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120597119871SVM120597120585119905

= 0 997888rarr 119862 minus 120572119905minus 120578119905= 0

120597119871SVM120597120585lowast

119905

= 0 997888rarr 119862 minus 120572lowast

119905minus 120578lowast

119905= 0

(5)

The dual problem can be obtained by substituting the rela-tions from (5)

maximize minus1

2

119873

sum

119905119897=1

(120572119905minus 120572lowast

119905) (120572119897minus 120572lowast

119897) ⟨120593 (119883

119905) 120593 (119883

119897)⟩

minus 120576

119873

sum

119905=1

(120572119905+ 120572lowast

119905) +

119873

sum

119905=1

119910119905(120572119905minus 120572lowast

119905)

subject to119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120572119905 120572lowast

119905isin [0 119862]

(6)

According to theKarush-Kuhn-Tucker (KKT) conditions theterms inside the equation containing the Lagrangemultipliers

will vanish at the optimal solution This gives the followingequations

120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887) = 0

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887) = 0

120578119905120585119905= (119862 minus 120572

119905) 120585119905= 0

120578lowast

119905120585lowast

119905= (119862 minus 120572

lowast

119905) 120585lowast

119905= 0

(7)

The above equations indicate that for all samples whoseLagrange multipliers are equal to zero are considered asnonsupport vectors Samples with nonzero coefficients areconsidered as support vectorsMeanwhile 119887 can be calculatedwhen the slack variables 120585

119905and 120585lowast119905are equal to zero

119887 = 119910119905minus ⟨119882 120593 (119883

119905)⟩ plusmn 120576 for 120572

119905 120572lowast

119905isin [0 119862] (8)

The final SVM for nonlinear functions can be written as

119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) ⟨120593 (119883) 120593 (119883

119905)⟩ + 119887 (9)

A modified version of (9) can be expressed as

119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905)119870 (119883119883

119905) + 119887 (10)

where119870(119883119883119905) = ⟨120593(119883) 120593(119883

119905)⟩ is called the kernel function

Gaussian radial basis kernel (11) is the most powerful one innonlinear function estimation

119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 119883

119905

1003817100381710038171003817

2

21205902

) (11)

22 Price Zones Analysis One of the special characteristicsof electricity is that it has to be produced as used Tak-ing PJM interconnected electricity market as an exampleelectricity price can vary from as low as ndash$10MWh to ashigh as $748MWh [30] The relationship between supplyand demand under normal deregulated market is no longerthe only dominant principle determining the electricityprice Other factors such as business competing strategyand unethical business behaviour are heavily involved andcannot be efficiently simulatedTherefore the accuracy of theforecasted electricity MCP in the peak price zone is alwaysvery low Although electricity MCP is very volatile it is stillnormally distributed along its average value [20ndash22] In theproposed work the electricity MCP is separated into fourprice zones low medium high and peak price zones Let120583 and 120590 be the mean and standard deviation of the monthlyhistorical electricityMCP the four price zones of eachmonthare determined based on the four criteria listed below

low MCP lt 120583 minus 120590medium 120583 minus 120590 le MCP lt 120583 + 05120590high 120583 + 05120590 le MCP lt 120583 + 15120590peak 120583 + 15120590 le MCP

Journal of Energy 5

About 15ndash25 of the price is in the low price zone50ndash60 of the price is in the medium price zone 15minus20 of the price is in the high price zone Finally the last5ndash10 of the price will be in the peak price zone Thestandard deviation multipliers are chosen based on the pricecharacteristics of the PJM interconnected electricity marketand forecasting model parameter optimization They can bemodified when applying in a different electricity marketInside the medium price zone most generating companiesare participating in supplying electricity Every participatinggenerating unit is running at its optimal output There isplenty of reserved energy available in the market which canbe supplied immediately Congestion is low in most areasand congestion cost is limited At this moment the marketis at its stable state Sudden shortage or surplus can be easilyrecovered or offset by other generating companies in the sameand nearby jurisdiction Supply and demand elasticity playsa major role and electricity MCP is mainly determined bythe demand and fuel cost Inside the low price zone theprofit margin is small and therefore only companies withlow production cost and mandatory all-time running powerunits such as nuclear power plants are participating Suddenshortage or surplus could cause price spikes because theycannot be recovered or offset as easily as in the medium pricezone In the peak price zone most power plants are runningat their peak capacity Transmission lines are under extremeload resulting in extremely high congestion cost in someareas System reserved energy is very low and generatingcompanies expect to take advantages and make huge profitsfrom it Electricity MCP is mainly determined by businesscompeting strategy and even unethical business behaviourin the peak price zone The high price zone is betweenthe medium and peak price zones and therefore showscharacteristics of both price zones The supply and demandrelationship and business competing strategy both dominatethe final electricity MCP inside the high price zone

23 Two-Stage Multiple SVM Forecasting Model ArchitectureThe architecture of the proposed two-stage multiple SVMbased midterm forecasting model of the electricity MCP isshown in Figure 2 It contains four layers (1) input layer (2)input data distribution layer (3) SVM price prediction layerand (4) output layer

Inside the input layer there will be input data such aselectricity hourly demand electricity daily peak demand anddaily natural gas price Data selection and preprocessing willbe explained in detail in Section 24 After all the data arepreprocessed inside the input layer a single SVM is firstutilized to predict the initial values of the electricity pricesThese price values are then separated into four predefinedprice zones low medium high and peak The input datadistribution layer can significantly reduce the range eachprice prediction SVM shall cover and improve the overallsystem accuracy when forecasting the midterm electricityMCP After the input data distribution is completed theSVM prediction module is utilized to forecast the midtermelectricity MCP There are four parallel connected SVMsinside the price prediction module to forecast the electricityMCP in four different price zones Based on the different

Inputdata

Peak priceSVM

High priceSVM

SVMinitial

forecasting

Pricezones

OutputseparationMedium price

SVM

Low priceSVM

X1

X2

Xn

Figure 2 Two-stage multiple SVM forecasting model architecture

range of input data each price prediction SVM optimizes itsown control parameters during the training process Thesedifferent control parameters give each price prediction SVMthe ability to capture the characteristics between input andoutput data within each price zone Inside the output layerfour SVMs predicted electricity price values are combined toform the final forecast electricity MCP Input data distribu-tion module will assign each input data with serial numbersthat will be used to regroup all forecast price values fromthe SVM price prediction layer inside the output layer Thetraining input data for both modules are from the inputlayer in Figure 2 The target data for both modules arethe historical electricity MCP Cross-validation technique isapplied in data distribution and price prediction modules forthe optimization of the control parameters

24 Data Collection and Preprocessing Electricity MCP canbe forecasted by evaluating a variety of elements [31] suchas electricity demand supply natural gas price coal pricehydrocapacity weather and temperature An ideal forecast-ing model of the electricity MCP should include all thepossible elements that affect the final electricity MCP Inreality however it is impossible and unnecessary to do soSince weather conditions including daily temperature arealready considered in load forecasting they do not have to beincluded in MCP forecasting process Other elements suchas business strategy and unethical competition behaviourscannot be easily represented mathematically Finally thecurrent failure or operating status of a generator in manyderegulated electric market is confidential information Dueto the factors mentioned above a data selection process ispresented in Figure 3 using cross-validation technique tofinalize the input data that were taken into account for theproposed work

The data selection process starts with only the defaultinput data (hourly electricity demand) inside the forecastingmodel Then the next element inside the potential inputdata is added to the initial forecasting model to test whetherthe added element improves the forecasting accuracy or notTested element will be added to the forecasting model if itimproves the forecasting accuracy Otherwise the forecastingmodel maintains its current input data and the next elementinside the potential input data will be tested Due to thefact that sometimes adding more than one element at a time

6 Journal of Energy

Yes

YesCross-validation on input data

selection is completed

Move on tothe next

combination

Potential input data

Forecasting model with hourlydemand Does the added input

data improve the forecastingaccuracy (yesno)

Move on tothe next

input dataNo

No

New input dataadded to the model

Is this the last possiblecombination of the potential

input data (yesno)

Figure 3 Data selection process

will achieve better forecasting results different combinationsof elements will also be tested once all the elements havebeen tested one at a time By applying the data selectionprocess shown in Figure 3 it will guarantee to test not onlyevery single element but also every possible combinationof elements that would lead to the optimized forecastingresults The final selected input data for the proposed two-stage multiple SVM forecasting model at each hour 119905 include(1) electricity hourly demand at hour 119905 (2) electricity dailypeak demand (3) electricity monthly average demand (4)daily price of natural gas (5) previous yearrsquos monthly averageelectricity MCP (6)month (1ndash12) and (7) hour of the day (1ndash24) The target data at hour 119905 is the electricity MCP at hour 119905Moreover one year is the most optimized length of historicaldata to train the forecasting model based on the previouspublished works [1ndash3] regarding the selection of training datafor midterm forecasting of the electricity MCP

Part of the training data is separated from the rest of thetraining data and served as the so-called the testing dataThe testing data are used to optimize the control parametersof the proposed forecasting model and are treated unseenfrom the rest of the training data For the rest of this paperthe remaining part of the training data is called the actualtraining data In the proposed work data from January 12009 to December 31 2009 excluding June 2009 are selectedas the actual training data and can be viewed as a matrixwith 8040 (24 times 335) rows and 8 columns The 8040 rowsare the number of hours from hour 1 (100 am) on January 12009 to hour 24 (1200 am) onDecember 31 2009 excludingJune 2009 The first 7 columns are the 7 input elementsmentioned beforeThe last column is the training target dataelectricity MCP from hour 1 (100 am) on January 1 2009to hour 24 (1200 am) on December 31 2009 excludingJune 2009 Because the proposed two-stage multiple SVMbased midterm forecasting model of the electricity MCP isdesigned to predict the 720 hourly electricity MCPs in June2010 selecting data from hour 1 (100 am) on June 1 2009 tohour 24 (1200 am) on June 30 2009 as the testing data couldcreate the best scenarios on daily demand pattern daily pricepattern weather sunshine and precipitation during trainingprocess to optimize the control parameters For the samereason if the proposed model is utilized to forecast the 744hourly electricity MCPs in May 2010 data in May 2009 will

be selected as the testing data during training process Thetesting data contain a matrix with 720 rows (24 times 30) and 8columns The first 7 columns are the testing input data andthe last column is the testing target data

Once the training procedure is done the proposed two-stage multiple SVM forecasting model is used to forecast themidterm electricity MCP of every hour in June 2010 usinghistorical data from January 2009 to December 2009 Thiswork is based on the assumption that all forecasting inputdata have already been accurately predicted to limit the affectsfrom inaccurate input data The forecast output contains720 (24 times 30) hourly electricity MCP which will insure thecomprehensiveness of the test sample The forecasting inputdata could be viewed as a matrix with 720 (24 times 30) rows and8 columnsThe 720 rows are the number of hours from hour 1(100 am) on June 1 2010 to hour 24 (1200 am) on June 302010 The first 7 columns are the forecasting input data andthe last column is the electricity MCP from hour 1 (100 am)on June 1 2010 to hour 24 (1200 am) on June 30 2010

In order to improve the regression accuracy and avoidthe dominance of some extremely large values inside the dataset data preprocessing is needed in machine learning Forelectricity MCP forecasting the most common and recom-mendeddata preprocessing algorithms are the normalizationNormalization converts each datum inside its correspondingelement into a value between ndash1 and +1 (sometimes between0 and 1) with respect to each elementrsquos maximum and mini-mumvalue Depending on the calculation themaximumandminimum values can be either global or local Suppose that119883119905= (1199091199051 1199091199052 119909

119905119896) is a given set of vectors at time 119905 with

119896 multiple elements for 119905 = 1 to 119873 The normalization fordatum 119909

119905119896could be defined as

119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)

where 119909MAX119896 and 119909MIN119896 are the global or local maximum andminimum values inside the 119896th element The normalizationon forecasting input data is performed using each elementrsquoscorresponding global or local maximum and minimumvalues during the training process The target data remainunchanged as there is only the electricity MCP data Thedesigned two-stagemultiple SVMbasedmidterm forecastingmodel of the electricity MCP can utilize different dimensionsof input data for forecasting accuracy optimization The fulldimension of the input data is 7 as mentioned before Recallthe data selection process from Figure 3 and using meanabsolute error for the performance evaluation the optimizedfinal dimensions of input data for the proposedmultiple SVMare 7 for the initial prediction SVM 5 for the low price SVM7 for the medium price SVM 7 for the high price SVM and7 for the peak price SVM

25 Performance Evaluation Mean absolute error (MAE)mean absolute percentage error (MAPE) and mean squareroot error (MSRE) are the three most widely used measure-ments for performance evaluation in forecasting electricityprice values Given119873 historical electricity MCP data 119910

119905 and

Journal of Energy 7

the corresponding forecasted price data 119910119905for 119905 = 1 to 119873

MAE MAPE and MSRE are defined as

MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)

During the training process two SVM control parametersthe coefficient of the Gaussian radial basis kernel 120590 andthe regularization constant 120574 are set at an initial value andupdated using testing data during each iteration until theMAE of the SVM reaches its global minimum Once allcontrol parameters are determined forecasting input data canbe applied to forecast the midterm electricity MCP

3 Case Studies

31 PJM Interconnected Electric Market PJM interconnectedelectric market is the largest interconnected system in theworld It includes all or most of Delaware District ofColumbia Maryland New Jersey Ohio Pennsylvania Vir-ginia and West Virginia Parts of Indiana Illinois KentuckyMichigan North Carolina and Tennessee are also includedin the PJM interconnected electric marketThemarket servesmore than 61 million people in the United States As ofDecember 31 2012 it had an installed generating capacity of183604 megawatts and more than 800 participating mem-bers The market covered 243417 square miles of territorycontaining 62556 miles of transmission lines and morethan 1376 electric generators [47] Coal (74) and naturalgas (22) are the two major types of fuel in the marketLocationalmarginal pricing (LMP) is used to reflect the valueof the energy at the specific location and time it is deliveredThemarket consists of day-ahead and real-time marketsTheday-ahead market is a forward market in which hourly LMPis calculated for the next operating day based on generationoffers demand bids and scheduled bilateral transactionsThereal-time market is a spot market in which current LMPis calculated at five-minute intervals based on actual gridoperating conditions PJM settles transactions hourly andissues invoices tomarket participantsmonthly [48]The LMPinside the PJM interconnected electric market is used aselectricity MCP in this work Historical data from the day-ahead market of 2008 and 2009 are used in this work toforecast the electricity MCP in June 2010 and are available at[47]

32 Experimental Results The testing results using bothsingle SVM and the proposed two-stage multiple SVM areshown in Figure 4 Data from January 2009 to December2009 excluding June 2009 are used as the actual training dataand data in June 2009 are used as the testing data

It can be seen that the major contribution of utilizinga two-stage multiple SVM based forecasting model is the

0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)

Real MCPSingle SVM forecast MCPTwo-stage multiple SVM forecast MCP

Figure 4 Forecasted electricityMCP in June 2009 using both singleand two-stage multiple SVM

Table 2 Performance evaluation results of MAE between the singleand the two-stage multiple SVM in June 2009

MAE Low Medium High Peak SystemSingle 40094 18941 30837 82403 29743Multiple 31985 18678 38336 57636 29625PRIM 2022 139 minus2432 3006 040

Table 3 Performance evaluation results of MAPE between thesingle and the two-stage multiple SVM in June 2009

MAPE Low Medium High Peak SystemSingle 359821 61526 72207 145750 117491Multiple 249639 56042 87352 112517 114083PRIM 3062 891 minus2097 2280 290

improved forecasting results in the low and peak price zonesThe regression computation performance evaluations areperformed using MAE MAPE andMSRE analysesThe per-formance numbers are shown in Tables 2 3 and 4 Percentageimprovement (PRIM) is used to compare the regressioncomputation performance between the single SVM modeland the proposed two-stage multiple SVMmodel The PRIMis calculated as

PRIM = ((single MAEMSREMAPE

minus multiple MAEMSREMAPE)

sdot (single MAEMSREMAPE)minus1) times 100

(14)

Positive PRIM values indicate that the two-stage multipleSVMmodel is better than the single SVMmodel in this pricezone Negative PRIM values indicate the opposite

According to all three tables MAE MAPE and MSREvalues show that the two-stage multiple SVM model is moreaccurate than the single SVM model by more than at least20 in the low and peak price zones However the results

8 Journal of Energy

Table 4 Performance evaluation results ofMSREbetween the singleand the two-stage multiple SVM in June 2009

MSRE Low Medium High Peak SystemSingle 05200 01191 02933 12738 01564Multiple 03381 01408 03590 10060 01536PRIM 3498 minus1822 minus2240 2102 179


MAE Low Medium High Peak SystemSingle 57523 69711 50316 159496 72523Multiple 27764 75398 76556 111233 68106PRIM 5173 minus816 minus5215 3026 609

are opposite in the high price zones This is caused by thecharacteristics of the high price zone as it is between themedium and peak price zones The supply and demandrelationship and business competing strategy both dominatethe final electricity MCP inside the high price zone It is verydifficult for a single SVM to simulate these two very differentcharacteristics at the same time Significantly reduced datacaused by the SVM classification process is also anotherreason MAE MAPE and MSRE are low in the high pricezone As the MSRE values in the medium price zones areopposite compared to the MAE and MAPE values it ishard to say which model has the better performance in themedium price zone Usually the conclusions obtained fromthe performance numbers ofMAEMAPE andMSRE shouldbe identical However in special cases the conclusions woulddiffer from each other due to the different distribution ofresults For instance suppose that the forecasting errors fromthe single SVM and two-stage multiple SVM are (6 6) and (110) and the forecasting percentage errors from the single SVMand two-stage multiple SVM are (6 6) and (1 10)The size of the data set is only 2 Therefore the MAE of thesingle SVM and the two-stage multiple SVM are (6 + 6)2 =6 and (1 + 10)2 = 55 According to the MAE values the two-stage multiple SVM is better than the single SVM The sameconclusions apply to theMAPE evaluation where theMAPEsof the single SVM and the two-stage multiple SVM are (6+ 6)2 = 6 and (1 + 10)2 = 55 However whencomparing the MSRE values the conclusion is different TheMSREs of the single SVM and the two-stage multiple SVMare (62 + 62)052 = 4243 and (12 + 102)052 = 5025 Accordingto the MSRE values the single SVM is better than the two-stage multiple SVM Under special situation like this whenthe conclusions obtained by the MAE MAPE and MSREvalues do not agree with each other in certain price zones(eg the medium zone) because all the control parametersare determined based on the global minimal MAE valueswe will use the MAE values to determine which forecastingmodel is better

As the testing results supported the fact that the two-stage multiple SVM is better than the single SVM inmidtermforecasting of the electricity MCP the proposed forecastingmodel is used to forecast the midterm hourly electricity


MAPE Low Medium High Peak SystemSingle 210094 163663 75146 167736 156454Multiple 102069 189028 124708 120534 146567PRIM 5142 minus1550 minus6595 2814 632



MCP in June 2010 using historical data from January 2009to December 2009 Figure 5 shows the forecast electricityhourly MCP in June 2010 using both single and the two-stagemultiple SVM The final midterm forecasting results of theelectricity hourly MCP are obtained from the combinationof predicted price from four price prediction SVMs Detailedevaluation results of MAE MAPE and MSRE are shown inTables 5 6 and 7 The two-stage multiple SVM forecastingmodel shows an average 6 improvement on the systemMAE MAPE and MSRE evaluations The negative PRIMvalues in the high price zone are expected while the negativePRIM values in the medium price zone indicate that there ismisdistribution during the forecast process The forecastingresults of the proposed multiple SVM model are also com-pared with other machine learning methods [43] shown inTable 8 Although the proposed two-stage multiple SVM isonly slightly better in forecasting accuracy compared withother midterm electricity MCP forecasting methods suchas the hybrid LSSVM and ARMAX during the forecastingprocess the proposed two-stage multiple SVM model isway more flexible in adjusting input data in any SVM andtherefore has much higher potential to improve in futureresearch than other midterm electricity price forecastingmethods All of the study cases were run on a PC with 4GBof RAM and a 24GHz processor MATLAB 77 is used Theaverage forecasting time is less than 1 minute

33 Discussions The SVM intends to lose the top and thebottom peak values during training process because most ofthose values are considered as nonsupport vectors when uti-lizing the 120576-insensitive loss function with 2120576 bandwidth Onlythe support vectors are used to create the 2120576 tube and only thevalues outside the 2120576 tube are considered during simulationAs mentioned before the electricity MCP in the peak pricezone is mainly determined by business competing strategyand even unethical business behaviour and those elementscannot be easily representedmathematicallyTherefore whenusing a single SVM forecasting model the forecast electricityMCP will lose most of its top and bottom peak values duringthe forecasting process The proposed two-stage multipleSVM forecasting model can improve the compromise thatis caused by using a single SVM by distributing the initially

Journal of Energy 9

Table 8 Performance evaluation results between the proposed two-stage multiple SVMmodel and other forecasting models

June 2009 June 2010MAE MAPE MSRE MAE MAPE MSRE

Single SVM 29743 117491 01564 72523 156454 03316Single LSSVM 28152 109722 01513 79003 162610 03964Hybrid LSSVM and ARMAX 27630 106706 01495 70989 139709 03859Multiple LSSVM 27811 107466 01463 70950 132850 03733Two-stage multiple SVM 29625 114083 01536 68106 146567 03126

20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)


Figure 5 Forecasted electricity MCP in June 2010 using both singleand two-stage multiple SVM

predicted price values into different price zones By doingso instead of having only two control parameters in a singleSVM forecasting model the proposed two-stage multipleSVM forecasting model has 10 control parameters that canbe adjusted to optimize the system forecasting accuracy

4 Conclusions

A two-stage multiple SVM based midterm forecasting modelof the electricity MCP is proposed in this paper Two mod-ules input data distribution module and price forecastingmodule are designed to first preprocess the initially predictedprice values into corresponding price zones and then forecastthe electricity price in four parallel designed SVMs Theproposed two-stage multiple SVM model showed improvedforecasting accuracy in the low and the peak price zones andthus improving the overall forecasting accuracy comparedto the forecasting model utilizing a single SVM The casestudies also show that the performance of either a singleor a two-stage multiple SVM is highly depending on theselection of the input data Carefully selected training inputdata and correctly predicted subdata sets would significantlyimprove the accuracy of the forecasting modelThe proposedforecasting model has ten adjustable control parameterscompared to only two adjustable control parameters in asingle SVM forecasting model The four parallel SVMs in

the price forecasting module also have the flexibility of usingdifferent dimensions of input data to achieve the optimalresults Future research will focus on possible modules thatcan be added to the distribution module to improve thesystem accuracy

Conflict of Interests

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

References

[1] S S Torbaghan A Motamedi H Zareipour and L A TuanldquoMedium-term electricity price forecastingrdquo in Proceedings ofthe North American Power Symposium (NAPS 12) pp 1ndash8Champaign Ill USA September 2012

[2] X Yan Electricity market clearing price forecasting in a deregu-lated electricity market [MS dissertation] Department of Elec-trical and Computer Engineering University of Saskatchewan2009

[3] X Yan and N A Chowdhury ldquoElectricity market clearing priceforecasting in a deregulated electricitymarketrdquo in Proceedings ofthe IEEE 11th International Conference on Probabilistic MethodsApplied to Power Systems (PMAPS rsquo10) pp 36ndash41 IEEE Singa-pore June 2010

[4] X Yan and N A Chowdhury Electricity Market ClearingPrice Forecasting in a Deregulated Market A Neural NetworkApproach VDM Saarbrucken Germany 2010

[5] S S Torghaban H Zareipour and L A Tuan ldquoMedium-termelectricity market price forecasting a data-driven approachrdquo inProceedings of the North American Power Symposium (NAPSrsquo10) pp 1ndash7 Arlington Va USA September 2010

[6] 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

[7] S J Yao and Y H Song ldquoPrediction of system marginal pricesby wavelet transform and neural networkrdquo in Proceedings of theElectric Machines and Power Systems pp 983ndash993 2000

[8] C-I Kim I-K Yu and Y H Song ldquoPrediction of systemmarginal price of electricity using wavelet transform analysisrdquoEnergy Conversion and Management vol 43 no 14 pp 1839ndash1851 2002

[9] J L Zhang and Z F Tan ldquoDay-ahead electricity price forecast-ing using WT CLSSVM and EGARCH modelrdquo InternationalJournal of Electrical Power and Energy Systems vol 45 no 1 pp362ndash368 2013

10 Journal of Energy

[10] C M Ruibal and M Mazumdar ldquoForecasting the mean andthe variance of electricity prices in deregulated marketsrdquo IEEETransactions on Power Systems vol 23 no 1 pp 25ndash32 2008

[11] Z Obradovic and K Tomsovic ldquoTime series methods forforecasting electricity market pricingrdquo in Proceedings of theIEEE Power Engineering Society Summer Meeting vol 2 pp1264ndash1265 Edmonton Canada 1999

[12] J Crespo Cuaresma J Hlouskova S Kossmeier and MObersteiner ldquoForecasting electricity spot-prices using linearunivariate time-series modelsrdquoApplied Energy vol 77 no 1 pp87ndash106 2004

[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 Gao X H Guan X-R Cao and A Papalexpoulos ldquoFore-casting powermarket clearing price and quantity using a neuralnetwork methodrdquo in Proceedings of the IEEE Power EngineeringSociety Summer Meeting vol 4 pp 2183ndash2188 Seattle WashUSA July 2000

[15] J P S Catalao S J P S Mariano V M F Mendes and L AF M Ferreira ldquoShort-term electricity prices forecasting in acompetitive market a neural network approachrdquo Electric PowerSystems Research vol 77 no 10 pp 1297ndash1304 2007

[16] P S Georgilakis ldquoMarket clearing price forecasting in dereg-ulated electricity markets using adaptively trained neural net-worksrdquo in Proceedings of the 4th Helenic Conference on ArtificialIntelligence pp 56ndash66 Heraklion Greece May 2006

[17] Z Xu Z Y Dong and W Liu ldquoNeural network models forelectricity market forecastingrdquo in Neural Networks Applicationsin Information Technology and Web Engineering D Wang andN K Lee Eds pp 233ndash245 Borneo Publishing CorporationSarawak Malaysia 1st edition 2005

[18] D Singhal and K S Swarup ldquoElectricity price forecasting usingartificial neural networksrdquo International Journal of ElectricalPower and Energy Systems vol 33 no 3 pp 550ndash555 2011

[19] S Anbazhagan and N Kumarappan ldquoDay-ahead deregulatedelectricity market price classification using neural networkinput featured byDCTrdquo International Journal of Electrical Powerand Energy Systems vol 37 no 1 pp 103ndash109 2012

[20] W Sun J-C Lu andMMeng ldquoApplication of time series basedSVM model on next-day electricity price forecasting underderegulated power marketrdquo in Proceedings of the InternationalConference on Machine Learning and Cybernetics pp 2373ndash2378 August 2006

[21] D M Zhou F Gao and X H Guan ldquoApplication of accurateonline support vector regression in energy price forecastrdquo inProceedings of the 5th World Congress on Intelligent Control andAutomation (WCICA rsquo04) pp 1838ndash1842 June 2004

[22] L L Hu G Taylor H BWan andM Irving ldquoA review of short-term electricity price forecasting techniques in deregulatedelectricity marketsrdquo in Proceedings of the 44th InternationalUniversities Power Engineering Conference pp 1ndash5 September2009

[23] R A Swief Y G Hegazy T S Abdel-Salam and M A BaderldquoSupport vector machines (SVM) based short term electricityload-price forecastingrdquo in Proceedings of the IEEE BucharestPowerTech pp 1ndash5 Bucharest Romania June 2009

[24] L M Saini S K Aggarwal and A Kumar ldquoParameter opti-misation using genetic algorithm for support vector machine-based price-forecasting model in National electricity marketrdquo

IET Generation Transmission amp Distribution vol 4 no 1 pp36ndash49 2010

[25] W Sun and J Zhang ldquoForecasting day ahead spot electricityprices based on GASVMrdquo in Proceedings of the InternationalConference on Internet Computing in Science and Engineering(ICICSE rsquo08) pp 73ndash78 Harbin China January 2008

[26] Y-G Chen and G W Ma ldquoElectricity price forecasting basedon support vector machine trained by genetic algorithmrdquo inProceedings of the 3rd International Symposium on IntelligentInformation Technology Application (IITA rsquo09) vol 2 pp 292ndash295 November 2009

[27] M J Mahjoob M Abdollahzade and R Zarringhalam ldquoGAbased optimized LS-SVM forecasting of short term electricityprice in competitive power marketsrdquo in Proceedings of the3rd IEEE Conference on Industrial Electronics and Applications(ICIEA rsquo08) pp 73ndash78 June 2008

[28] D-S Gong J-X Che J-Z Wang and J-Z Liang ldquoShort-termelectricity price forecasting based on novel SVM using artificialfish swarm algorithm under deregulated powerrdquo in Proceedingsof the 2nd International Symposium on Intelligent InformationTechnology Application pp 85ndash89 December 2008

[29] H Zheng L Z Zhang L Xie and X Li ldquoSVMmodel of systemmarginal price based on developed independent componentanalysisrdquo in Proceedings of the International Conference onPower System Technology pp 1437ndash1440 2004

[30] Y Wang and S Q Yu ldquoPrice forecasting by ICA-SVM in thecompetitive electricity marketrdquo in Proceedings of the 3rd IEEEConference on Industrial Electronics and Applications (ICIEArsquo08) pp 314ndash319 Singapore June 2008

[31] J Y Tian and Y Lin ldquoShort-term electricity price forecastingbased on rough sets and improved SVMrdquo in Proceedings of the2nd International Workshop on Knowledge Discovery and DataMining (WKKD rsquo09) pp 68ndash71 IEEE Moscow Russia January2009

[32] T Wang and L Qin ldquoApplication of SVM based on rough setin electricity prices forecastingrdquo in Proceedings of the 2nd Con-ference on Environmental Science and Information ApplicationTechnology (ESIAT rsquo10) pp 317ndash320 IEEE Wuhan China July2010

[33] L Xie H Zheng and L Z Zhang ldquoElectricity price forecastingby clustering-LSSVMrdquo in Proceedings of the 8th InternationalPower Engineering Conference pp 697ndash702 December 2007

[34] S Fan J R Liao K Kaneko and L N Chen ldquoAn integratedmachine learning model for day-ahead electricity price fore-castingrdquo in Proceedings of the IEEE PES Power Systems Confer-ence and Exposition (PSCE rsquo06) pp 1643ndash1649 November 2006

[35] S Fan C Mao and L Chen ldquoNext-day electricity-price fore-casting using a hybrid networkrdquo IET Generation Transmissionand Distribution vol 1 no 1 pp 176ndash182 2007

[36] J H Zhao Z Y Dong Z Xu and K P Wong ldquoA statisticalapproach for interval forecasting of the electricity pricerdquo IEEETransactions on Power Systems vol 23 no 2 pp 267ndash276 2008

[37] J H Zhao Z Y Dong X Li and K PWong ldquoA general methodfor electricity market price spike analysisrdquo in Proceedings of theIEEE Power Engineering Society General Meeting pp 286ndash293June 2005

[38] J H Zhao Z Y Dong X Li and K P Wong ldquoA frameworkfor electricity price spike analysis with advanced data miningmethodsrdquo IEEE Transactions on Power Systems vol 22 no 1pp 376ndash385 2007

Journal of Energy 11

[39] P Areekul T Senjyu H Toyama and A Yona ldquoA hybridARIMA and neural network model for short-term price fore-casting in deregulated marketrdquo IEEE Transactions on PowerSystems vol 25 no 1 pp 524ndash530 2010

[40] L Xie L Z Zhang H Zheng and Y Xie ldquoOn the sensitivityanalysis of electricity pricerdquo in Proceedings of the 2nd IEEEConference on Industrial Electronics and Applications pp 1603ndash1606 May 2007

[41] Y Wang ldquoPrice-load elasticity analysis by hybrid algorithmrdquo inProceedings of the 3rd International Conference on Electric UtilityDeregulation and Restructuring and Power Technologies (DRPTrsquo08) pp 233ndash237 IEEE Nanjing China April 2008

[42] H M I Pousinho V M F Mendes and J P S Catalao ldquoShort-term electricity prices forecasting in a competitive marketby a hybrid PSO-ANFIS approachrdquo International Journal ofElectrical Power and Energy Systems vol 39 no 1 pp 29ndash352012

[43] X Yan and N A Chowdhury ldquoMid-term electricity marketclearing price forecasting a hybrid LSSVM and ARMAXapproachrdquo International Journal of Electrical Power amp EnergySystems vol 53 no 1 pp 20ndash26 2013

[44] D J Pedregal and J R Trapero ldquoMid-term hourly electricityforecasting based on a multi-rate approachrdquo Energy Conversionand Management vol 51 no 1 pp 105ndash111 2010

[45] J G Gonzalez J Barquın and P Duenas ldquoA hybrid approachfor modeling the electricity price series in the medium termrdquoin Proceedings of the Power Systems Computation Conference(PSCC rsquo08) 2008

[46] C Cortes and V Vapnik Support-Vector Networks ATampT BellLabs Holmdel NJ USA 1995

[47] Electricity LMP in PJM 2014 httpwwwpjmcommarkets-and-operationsenergyday-aheadlmpdaaspx

[48] Federal Energy Regulatory Commission 2010 httpwwwfercgov

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of


Journal ofPetroleum Engineering


Industrial EngineeringJournal of


Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of


Journal of


Renewable Energy

Submit your manuscripts athttpwwwhindawicom


StructuresJournal of


RotatingMachinery


EnergyJournal of


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Nuclear InstallationsScience and Technology of


Solar EnergyJournal of


Wind EnergyJournal of


Nuclear EnergyInternational Journal of


High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

2 Journal of Energy

past locating the peak prices in most cases is usually veryaccurateThemain challenge in short-term forecasting is howaccurately the forecasting model can predict the values ateach peak price spot However in the midterm forecastingof the electricity MCP because of the unavailability of datafrom the immediate past locating the peak prices becomesextremely difficult As a result accurately predicting thevalues at peak price spots becomes even more difficultThe fourth difference is the length of historical data thatare needed to train the forecasting model The short-termforecasting of the electricity price model usually only needsthe data of the last couple of days to train the forecastingmodel On the other hand the midterm forecasting modelrequires one year of historical data in order to train theforecasting model Finally only nonlinear regression basedforecastingmodels are capable ofmidterm forecasting Linearregression based forecasting models can only be consideredas an add-on module in the midterm forecasting

Before hybrid models utilized in the forecasting of theelectricity MCP autoregressive integrated moving average(ARIMA) [6] wavelet transform [7ndash9] Monte Carlo sim-ulation [10] bid-based stochastic model [10] time series[11 12] and dynamic regression [13] are the first generationof techniques utilized in the forecasting of the electricityMCP Artificial neural network (ANN) was later on appliedto forecast short-term electricity MCP [2ndash4 14ndash19] becauseof its flexibility in handling highly nonlinear relationshipsand relatively easy implementation PJM interconnectionAustralian electricity market England-Wales pool and NewEngland ISO [16] are currently utilizing ANN to forecast theelectricity MCP

Support vector machine (SVM) is a new learningmethodbased on structural risk minimization and has gainedincreased attention in recent electricity price forecasting [20ndash23] studies The major advantages of SVM over ANN or anyother forecasting models are that SVM can avoid problemssuch as data overfitting local minimum and unpredictablylarge out-of-sample data error while at the same time stillachieving better results SVM is also a very robust forecastingmodel and will always end up with the same acceptableresult regardless of the initial values On top of that SVMhas less adjustable parameters compared to ANN and isless complicated in control parameter selection In electricityprice forecasting a traditional SVM can achieve around 3[24] improved forecasting accuracy compared to a traditionalANN Several algorithms including genetic algorithm (GA)[24ndash27] artificial fish swarm algorithm [28] independentcomponent analysis (ICA) algorithm [29 30] and rough setsalgorithm [31 32] are used to further optimize the trainingof SVM Moreover the least squares support vector machine(LSSVM) was later developed to improve the accuracy of theoriginal SVM [25 27 33ndash35] Although each method hasshown some improvements the overall system accuracy wasstill quite low

The new trend in recent short-term forecasting of theelectricity price studies is using hybrid models combiningseveral prediction methods Hybrid models can compensatetheweaknesses of utilizing any individual establishedmethodand achieve better overall system results Torghaban et al [5]

proposed hybrid midterm electricity monthly average priceforecasting models combined with SVMSVM SVMNNNNSVM and NNNN Swief et al [23] utilized the principlecomponent analysis (PCA) and 119896 nearest neighbor (119896NN)points technique to reduce the number of data entries to theSVM model Wang and Qin [32] introduced an electricityprice prediction model using rough sets and SVM wherethe rough sets theory was used to simplify the SVM inputvariables Xie et al [33] proposed a hybrid electricity priceforecasting model that integrates clustering algorithm withLSSVM Fan et al [35] proposed a hybrid forecasting modelthat includes the Bayesian clustering by dynamics (BCD)and SVM where the BCD classifier is applied to clusterthe input data set into several subsets in an unsupervisedmanner Zhao et al [36] forecasted the prediction intervalof the electricity price utilizing SVM and nonlinear con-ditional heteroscedastic forecasting (NCHF) model Theyalso proposed a hybrid model [37 38] using SVM andprobability classifier to predict the price spike occurrenceand values Areekul et al [39] proposed forecast techniquesbased on autoregressive integratedmoving average (ARIMA)and ANN Xie et al [40] proposed a sensitivity analysis ofelectricity price utilizing SVM and regression model whereseveral price-load elasticity equations are built based on theprice pattern data sets classified by SVM classification Asimilar work focused on price-load elasticity analysis is doneby Wang in [41] Pousinho et al [42] proposed a hybridshort-term electricity price forecastingmodel combined withparticle swarm optimization (PSO) and adaptive-networkbased fuzzy inference system (ANFIS)

A new midterm forecasting model of the electricityMCP utilizing two-stage multiple SVM is proposed in thispaper based on the studying of previous works in midtermforecasting of the electricity MCP shown in Table 1 Theproposed model can predict both patterns and values ofelectricity MCP A single SVM is first used to separate theinput data into corresponding price zonesThen four paralleldesigned SVMs are used to forecast the price values insideeach separated price zone The main contributions of thispaper include (1) addressing and resolving problems asso-ciated with midterm forecasting of the electricity MCP (2)addressing and resolving the problems associated with usinga single nonlinear forecasting model to forecast midtermelectricity MCP such as SVM and (3) presenting a two-stage multiple SVM based forecasting model Experimentalresults using historical data from the PJM interconnectionsystem demonstrated that the proposed two-stage multipleSVM forecasting model can improve the price predictingaccuracy in both low and peak price zones The rest ofthe paper is organized as follows Two-stage multiple SVMbased midterm forecasting model of the electricity MCP isgiven in Section 2 Case studies are presented in Section 3Conclusions are included in Section 4

2 Two-Stage Multiple SVM Based MidtermForecasting Model of the Electricity MCP

21 Support Vector Machine SVM was first introduced byVladimir Vapnik in 1979 based on the statistical learning and

Journal of Energy 3








peak prices












Gonzalez et al [45]




X = [x1 x2 xN]

x1

x2

xN

N inputs

sum

Bias

Output neuron

Hidden layer

K(x xi)

K(x xi)

K(x xi)






data where119883119905= (1199091199051 1199091199052 119909




119910119905= 119891 (119883

119905) = ⟨119882 120593 (119883

119905)⟩ + 119887 (1)




minimize 1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

subject to 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887 le 120576 + 120585

119905

⟨119882 120593 (119883119905)⟩ + 119887 minus 119910

119905le 120576 + 120585

lowast

119905

120585119905 120585lowast

119905ge 0

(2)




4 Journal of Energy


1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816120576=

0 if 1003816100381610038161003816119910119905 minus 119891 (119883119905)1003816100381610038161003816le 120576

1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816minus 120576 otherwise

(3)



=1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

minus

119873

sum

119905=1

120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887)

minus

119873

sum

119905=1

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887)

minus

119873

sum

119905=1

(120578119905120585119905+ 120578lowast

119905120585lowast

119905)

(4)




119905 and 120585lowast

119905) are then

120597119871SVM120597119882

= 0 997888rarr 119882 =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) 120593 (119883

119905)

120597119871SVM120597119887

= 0 997888rarr

119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120597119871SVM120597120585119905

= 0 997888rarr 119862 minus 120572119905minus 120578119905= 0

120597119871SVM120597120585lowast

119905



119905= 0

(5)


maximize minus1

2

119873

sum

119905119897=1

(120572119905minus 120572lowast

119905) (120572119897minus 120572lowast

119897) ⟨120593 (119883

119905) 120593 (119883

119897)⟩

minus 120576

119873

sum

119905=1

(120572119905+ 120572lowast

119905) +

119873

sum

119905=1

119910119905(120572119905minus 120572lowast

119905)

subject to119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120572119905 120572lowast

119905isin [0 119862]

(6)



120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887) = 0

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887) = 0

120578119905120585119905= (119862 minus 120572

119905) 120585119905= 0

120578lowast

119905120585lowast

119905= (119862 minus 120572

lowast

119905) 120585lowast

119905= 0

(7)



119887 = 119910119905minus ⟨119882 120593 (119883

119905)⟩ plusmn 120576 for 120572

119905 120572lowast

119905isin [0 119862] (8)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) ⟨120593 (119883) 120593 (119883

119905)⟩ + 119887 (9)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905)119870 (119883119883

119905) + 119887 (10)

where119870(119883119883119905) = ⟨120593(119883) 120593(119883



119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 119883

119905

1003817100381710038171003817

2

21205902

) (11)



Journal of Energy 5




Inputdata

Peak priceSVM

High priceSVM

SVMinitial

forecasting

Pricezones


SVM

Low priceSVM

X1

X2

Xn





6 Journal of Energy

Yes



Move on tothe next

combination




Move on tothe next

input dataNo

No



input data (yesno)










119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)



119905 and

Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 3








peak prices












Gonzalez et al [45]




X = [x1 x2 xN]

x1

x2

xN

N inputs

sum

Bias

Output neuron

Hidden layer

K(x xi)

K(x xi)

K(x xi)






data where119883119905= (1199091199051 1199091199052 119909




119910119905= 119891 (119883

119905) = ⟨119882 120593 (119883

119905)⟩ + 119887 (1)




minimize 1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

subject to 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887 le 120576 + 120585

119905

⟨119882 120593 (119883119905)⟩ + 119887 minus 119910

119905le 120576 + 120585

lowast

119905

120585119905 120585lowast

119905ge 0

(2)




4 Journal of Energy


1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816120576=

0 if 1003816100381610038161003816119910119905 minus 119891 (119883119905)1003816100381610038161003816le 120576

1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816minus 120576 otherwise

(3)



=1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

minus

119873

sum

119905=1

120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887)

minus

119873

sum

119905=1

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887)

minus

119873

sum

119905=1

(120578119905120585119905+ 120578lowast

119905120585lowast

119905)

(4)




119905 and 120585lowast

119905) are then

120597119871SVM120597119882

= 0 997888rarr 119882 =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) 120593 (119883

119905)

120597119871SVM120597119887

= 0 997888rarr

119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120597119871SVM120597120585119905

= 0 997888rarr 119862 minus 120572119905minus 120578119905= 0

120597119871SVM120597120585lowast

119905



119905= 0

(5)


maximize minus1

2

119873

sum

119905119897=1

(120572119905minus 120572lowast

119905) (120572119897minus 120572lowast

119897) ⟨120593 (119883

119905) 120593 (119883

119897)⟩

minus 120576

119873

sum

119905=1

(120572119905+ 120572lowast

119905) +

119873

sum

119905=1

119910119905(120572119905minus 120572lowast

119905)

subject to119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120572119905 120572lowast

119905isin [0 119862]

(6)



120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887) = 0

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887) = 0

120578119905120585119905= (119862 minus 120572

119905) 120585119905= 0

120578lowast

119905120585lowast

119905= (119862 minus 120572

lowast

119905) 120585lowast

119905= 0

(7)



119887 = 119910119905minus ⟨119882 120593 (119883

119905)⟩ plusmn 120576 for 120572

119905 120572lowast

119905isin [0 119862] (8)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) ⟨120593 (119883) 120593 (119883

119905)⟩ + 119887 (9)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905)119870 (119883119883

119905) + 119887 (10)

where119870(119883119883119905) = ⟨120593(119883) 120593(119883



119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 119883

119905

1003817100381710038171003817

2

21205902

) (11)



Journal of Energy 5




Inputdata

Peak priceSVM

High priceSVM

SVMinitial

forecasting

Pricezones


SVM

Low priceSVM

X1

X2

Xn





6 Journal of Energy

Yes



Move on tothe next

combination




Move on tothe next

input dataNo

No



input data (yesno)










119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)



119905 and

Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















4 Journal of Energy


1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816120576=

0 if 1003816100381610038161003816119910119905 minus 119891 (119883119905)1003816100381610038161003816le 120576

1003816100381610038161003816119910119905minus 119891 (119883

119905)1003816100381610038161003816minus 120576 otherwise

(3)



=1

21198822+ 119862

119873

sum

119905=1

(120585119905+ 120585lowast

119905)

minus

119873

sum

119905=1

120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887)

minus

119873

sum

119905=1

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887)

minus

119873

sum

119905=1

(120578119905120585119905+ 120578lowast

119905120585lowast

119905)

(4)




119905 and 120585lowast

119905) are then

120597119871SVM120597119882

= 0 997888rarr 119882 =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) 120593 (119883

119905)

120597119871SVM120597119887

= 0 997888rarr

119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120597119871SVM120597120585119905

= 0 997888rarr 119862 minus 120572119905minus 120578119905= 0

120597119871SVM120597120585lowast

119905



119905= 0

(5)


maximize minus1

2

119873

sum

119905119897=1

(120572119905minus 120572lowast

119905) (120572119897minus 120572lowast

119897) ⟨120593 (119883

119905) 120593 (119883

119897)⟩

minus 120576

119873

sum

119905=1

(120572119905+ 120572lowast

119905) +

119873

sum

119905=1

119910119905(120572119905minus 120572lowast

119905)

subject to119873

sum

119905=1

(120572119905minus 120572lowast

119905) = 0

120572119905 120572lowast

119905isin [0 119862]

(6)



120572119905(120576 + 120585

119905minus 119910119905+ ⟨119882 120593 (119883

119905)⟩ + 119887) = 0

120572lowast

119905(120576 + 120585

lowast

119905+ 119910119905minus ⟨119882 120593 (119883

119905)⟩ minus 119887) = 0

120578119905120585119905= (119862 minus 120572

119905) 120585119905= 0

120578lowast

119905120585lowast

119905= (119862 minus 120572

lowast

119905) 120585lowast

119905= 0

(7)



119887 = 119910119905minus ⟨119882 120593 (119883

119905)⟩ plusmn 120576 for 120572

119905 120572lowast

119905isin [0 119862] (8)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905) ⟨120593 (119883) 120593 (119883

119905)⟩ + 119887 (9)


119910119905= 119891 (119883

119905) =

119873

sum

119905=1

(120572119905minus 120572lowast

119905)119870 (119883119883

119905) + 119887 (10)

where119870(119883119883119905) = ⟨120593(119883) 120593(119883



119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 119883

119905

1003817100381710038171003817

2

21205902

) (11)



Journal of Energy 5




Inputdata

Peak priceSVM

High priceSVM

SVMinitial

forecasting

Pricezones


SVM

Low priceSVM

X1

X2

Xn





6 Journal of Energy

Yes



Move on tothe next

combination




Move on tothe next

input dataNo

No



input data (yesno)










119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)



119905 and

Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 5




Inputdata

Peak priceSVM

High priceSVM

SVMinitial

forecasting

Pricezones


SVM

Low priceSVM

X1

X2

Xn





6 Journal of Energy

Yes



Move on tothe next

combination




Move on tothe next

input dataNo

No



input data (yesno)










119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)



119905 and

Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















6 Journal of Energy

Yes



Move on tothe next

combination




Move on tothe next

input dataNo

No



input data (yesno)










119909119905119896=

119909119905119896minus (119909MAX119896 + 119909MIN119896) 2

119909MAX119896 minus (119909MAX119896 + 119909MIN119896) 2 (12)



119905 and

Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 7



MAE = 1

119873

119873

sum

119905=1

1003816100381610038161003816119910119905minus 119910119905

1003816100381610038161003816

MAPE = 1

119873

119873

sum

119905=1

10038161003816100381610038161003816100381610038161003816

119910119905minus 119910119905

119910119905

10038161003816100381610038161003816100381610038161003816

times 100

MSRE = radic 1

119873

119873

sum

119905=1

(119910119905minus 119910119905)2

(13)


3 Case Studies




0 100 200 300 400 500 600 700

10

20

30

40

50

60

70

80

Hour

Elec

tric

ity M

CP ($

MW

h)











(14)



8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















8 Journal of Energy













Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Journal of Energy 9




20

40

60

80

100

120

140

0 100 200 300 400 500 600 700Hour

Elec

tric

ity M

CP ($

MW

h)




4 Conclusions





References
























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















research article midterm electricity market clearing price … · 2018. 11. 30. · research...

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