Predictability of price movements in deregulated electricity markets

4 MB·English

by  Olga Y. Uritskaya

Predictability of price movements in deregulated electricity markets
OlgaY.Uritskaya
QuantitativeDynamicsLLC,2205DarrowSt,SilverSpringMD,USA
VadimM.Uritsky
CatholicUniversityofAmerica,620MichiganAveNE,WashingtonDC,USA
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1
0 Abstract
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InthispaperweinvestigatepredictabilityofelectricitypricesintheCanadianprovincesofAlbertaandOntario,
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a aswellasintheUSMid-Cmarket. Usingscale-dependentdetrendedfluctuationanalysis,spectralanalysis,andthe
J probabilitydistributionanalysisweshowthatthestudiedmarketsexhibitstronglyanti-persistentpropertiessuggesting

3 thattheirdynamicscanbepredictedbasedonhistoricpricerecordsacrosstherangeoftimescalesfromonehourto
2 one month. For both Canadian markets, the price movements reveal three types of correlated behavior which can
beusedforforecasting. Thediscoveredscenariosremainthesameondifferenttimescalesuptoonemonthaswell
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T as for on- and off- peak electricity data. These scenarios representsharp increases of prices and are not presentin
S the Mid-C marketdue to its lower volatility. We arguethat extremeprice movementsin this market shouldfollow
. the same tendency as the more volatile Canadian markets. The estimated values of the Pareto indices suggest that
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the predictionof these events can be statistically stable. The results obtainedprovidenew relevantinformationfor
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f managingfinancialrisksassociatedwiththedynamicsofelectricityderivativesovertimeframeexceedingoneday.
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[ Keywords:
Deregulatedelectricitymarkets,efficientmarkethypothesis,detrendedfluctuationanalysis,financialforecasting
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v
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1 1. Introduction
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8 Themodernelectricitymarketisnotonlyasystemforarrangingthepurchaseandsaleofelectricityusingsupply
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anddemandtosettheprice,but,formostmajorgrids,isabasisforelectricityderivatives,suchaselectricityfutures
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5 and options, which are actively traded. The practical significance of this part of the market is increasing as is the
0 importanceoftherelatedscientificresearch[1,2,3]. Themarketsofelectricityderivativeshavedevelopedasaresult
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oftheliberalizationandderegulationofelectricpowersystemsaroundtheworld.Deregulation,introducedinitiallyto
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reduceandsimplifythecontrolofthebusinessinthisfield,hadafinalgoaltoreachfinancialefficiencyofelectricity
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v markets[4, 5]. However,electricity is uniqueasit is a non-storablecommodity,andthe marketsremainextremely
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X inefficient[6,7].
Electricity prices are not a result of long-termbut instant, usually on an hourly interval, balance of supply and
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a demand. Moreover,asaconsequenceofthecomplexityofawholesaleelectricitymarket,itcanshowanextremely
highpricevolatilityattimesofpeakdemandandsupplyshortages.Thispricespikesarehardtopredictandfinancial
riskmanagementisstillahighpriorityforparticipantsinderegulatedelectricitymarketsduetothesubstantialprice
andvolumerisksthatthemarketscanexhibit[8,9,10].
Theproblemofpredictabilityofelectricitypricesinderegulatedmarketshasbeenconsideredinmanyprevious
studies(forinstance,[11,12,13,14]).Thevaluesofpricescanvarybyafactorof100overatimescaleofjustseveral
hours. These dramatic changes tend to occur in a seemingly spontaneousfashion which is sometimes erroneously
interpretedas a signature of a random uncorrelatedprocess (see for example [15]). A more detailed mathematical
analysisrevealsnontrivialauto-correlationsinthesesuddenpricejumps[16,17,18,19]whichindicateapossibility
ofpredictionofelectricitypricemovementsbasedontheinformationontheirhistoricevolution[7]. However,itisa
widelyrecognizedfactthatpricefluctuationsinenergymarketsdisplayheavydistributiontails[16,17,20,21,22,23]
PreprintsubmittedtoElsevier June1,2015
causingsubstantialdifficultiesinbuildingquantitativeforecastingmodelsofpricebehavior. Lessattentionhasbeen
paidto theanalysisoftemporalpatternsunderlyingthe observedstatistical structureofelectricitymarketsandasa
resultmodelingoftheirdynamicsisstillinitsinfancyandistypicallylimitedtoday-aheadmodels[2,24,25,26,27].
In this study, we take a few next steps toward answering fundamentalquestions related to the predictability of
electricity prices. First of all, can deregulatedelectricity markets reach the state of efficiency with the Hurst index
valueclosetootherwell-knownmarkets,orthisstateisnotreachableintheusualsense[28,29,30]?Thisiscrucially
importantbecauseiftheelectricitymarketsareinherentlyinefficient,theforecastcanbebuiltatvarioustimescales.
Inthiscontext,theinefficiencymeansthatpricehistoryisrelevanttothefuturepricechangesandcanbeusedfortheir
forecasting[31]. TheproblemwithPareto-typestatisticsplaysaspecialpart,becausenotallheavydistributiontails
canbeapproximatedbyasingleprobabilisticmodel. Theycanincludeseveraldynamicrangesdescribedbydistinct
Paretoexponents. Ifsuchmarketsarepredictableinprinciple,theremightbeparticularpriceintervalsforwhichthe
forecastisstatisticallystable,andtheseintervalsareimportanttoidentify.
Another central question related to the predictability of electricity prices is how universal can be a model of
electricity price behavior across different markets. In the present work, this question is addressed in frames of a
quantitativeanalysisofelectricitypricesinthreeindependentmarketswithdifferentlevelsofliberalization–Alberta,
Ontario(Canada)andMid-C(USA)markets.
Dynamicalandstatisticalpropertiesofpricefluctuationsareinvestigatedusingseveralmethods.First,weevaluate
correlationsinpricedynamicsacrossdifferenttimescalesusingthemethodofscale-dependentfractalexponent[7]
obtained from detrended fluctuation analysis (DFA) [32, 33, 34, 35]. We also use the Fourier spectral analysis to
identifycycliccomponentsintheelectricitypricedynamics,aswellastheParetoprobabilitydistributionanalysisfor
testing the stability of statistical momentsof the studied data. Spectraland DFA analysisresultsshow no evidence
ofinformationalefficiencyofelectricitypricefluctuationsatanytimescale. Allthreemarketsdemonstratedifferent
levelsofinefficiencywhichcouldreflecttheirdifferentsizesandstructuraldiversification. Pricemovementsinthese
markets are strongly anti-persistent [28]. Together with Pareto analysis results, this anti-persistence indicates that
electricitypricemovementscanbepredictedbasedonhistoricpricerecords.
Next,weverifythepossibilityofpriceforecastingusingphasediagramsrepresentingthecorrelationofprevious
and current price increments. According to our results, the diagrams have a complex asymmetric shape revealing
three basic scenarios of price movements. These scenarios remain the same for price movementsat different time
scales,fromonehouruptoonemonth,andarefoundtorepresentstronglyvolatilemarketconditions.Basedonthese
results,weshowthatpricefluctuationsinderegulatedelectricitymarketsarepredictablebytheirnature.Ourfindings
layafoundationforfuturemathematicaldescriptionofmultiscaledynamicsinderegulatedelectricitymarkets.
Theplanofthepaperisthefollowing.Section2containsadetaileddescriptionoftheanalyzeddatasets. Section
3describesmainresultsofourstatisticalanalysisdemonstratingthepossibilityofelectricitypriceforecasting. This
possibilityisexploredfurtherinSections4. Section5providesabriefsummaryofourstudy.
2. Data
Asanoutcomeoftheliberalizationpoliciespursuedinseveralcountriesfromthe80son,thesocalledday-ahead
electricity market provides economists with a very challenging phenomenon. Electricity cannot be economically
stored,whichimpliesthatdemandandsupplymustbecontinuouslybalanced,sothatthemarketpricemainlyreflects
the demandand supply conditionsprevailingat the very momentit has to be delivered to final users. Then, rather
complexmarketsystemshavebeensetup,withtheaimofreachingareasonabletrade-offbetweeneconomicefficiency
and system reliability. These systems are built around a market operator, whose task is to manage uniform-price,
sealed-bid,bilateralauctionsinordertoconstructaggregatedemandandsupplycurves,andtodetermineequilibrium
pricesandquantities. Theknife-edgecharacterofsuchapricesettingmechanismisfatherlypushedtotheextreme
byaverylowpriceelasticityofdemand,andbytechnicalconstraintswhichtimebytimeleadtonetworkcongestion
(seee.g. [23,36]andreferencestherein).
ThedatastudiedinthispaperconsistsofhourlyrealtimepoolelectricitypricesinAlberta,postedbytheAlberta
ElectricSystemOperator(AESO),andOntario,postedbytheIndependentElectricitySystemOperator(IESO).The
datacovertheperiodfromMay1,2002toJune6, 2009. InadditiontotheseCanadianmarkets,theMidColumbia
(Mid-C) market has been considered during the time interval 1 July, 2001 to 31 Oct, 2006. For each of the three
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Figure1: Timeseriesofhourlyelectricity pricesinAlberta(left), Ontario(center)andMid-C(right)markets. Fromtoptobottom: allhourly
prices,on-peakprices,andoff-peakprices. AlbertaelectricitypricesdemonstratesignificantlyhigherfluctuationsthanthoseinOntariomarket,
plottedonthesameverticalscale. FluctuationsofelectricitypricesinMid-ChavetwiceaslowamplitudeasthatinOntario,andabout5times
smallerthaninAlberta.
hourlydatasets, two secondarytime seriesconsistingof electricitypricesduringon-andoff-peakhourshavealso
beenexamined.Figure1showsthetimeseriesunderstudy,includingtheoriginaldataandtheiron-andoff-subsets.
Notethatallplotscontainnumerousspikeswithirregulartimingandamplitude.
AlbertaandOntarioaretheonlytwoCanadianprovinceswherewholesaleelectricitymarketsarefullyderegulated
[13,37].Alberta’smarketisdominatedbyfossilfuelgenerationandassuchfollowsmorecloselythepriceofnatural
gas. Ontario’s generationinvolvesabout50% of nuclear and 25% of hydropower [38, 39] enabling a more stable
price behavior [40]. The average level of volatility of electricity prices in Alberta is about twice as high as in the
Ontariomarket.
TheMidColumbiaelectricitymarketisnotasderegulatedasAlbertaandOntarioare[41]. Itisnotacentralized
powermarket,butitisatradinghubwherepowerisbilaterallytradedamongutilitiesandmarketers.TheMid-Cprice
hubisareferencepriceforthePacificNWregion,whichconsistsofWashington,Idaho,andOregon. Inthisregion,
largeutilitiesowngenerationandserveloadunderregulatedrates. Thegenerationisprimarilyhydroandtheregion
typically exportsto British Columbia and California [7, 42]. For these reasons, Mid-C prices are significantly less
volatilethanthoseineitherCanadianmarket.
3. Statisticalsignaturesandpredictability
3.1. DFAandspectralsignatures
Fortestingtheinformationalefficiencyofelectricitypricefluctuations,multiscalecorrelationsofpricedynamics
were evaluated across different time scales. Two complementary approaches were used to achieve this goal – the
scale-dependentDFAandtheFourierspectralanalysis.
Theformerofthetwoapproacheshasbeenfirstintroducedin[7]. Incontrasttopreviousmethodsmanipulating
with average scaling exponents characterizing broad scaling ranges, we investigated the distribution of local DFA
exponentsover all time scales involved. This approach was shown to be the only suitable when the signals under
study encompass qualitatively different types of behavior including random price movements, cycles, and spikes.
UsingtheDFAasthebasealgorithmisjustifiedbythepresenceofmultiscaletrendsintheelectricitydata[29,43].
Thescale-dependentversionofthisalgorithmpresentedin[7]enablestheinvestigationofcomplextypesofnonlinear
behavior of financial and economic indicators by providingdetailed informationon the distribution of correlations
overdifferentscales,andisespeciallyusefulforquantitativeanalysisofmarketefficiency.
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Figure2:DependenceofthedetrendedvariationF(eq. (2))andthelocalscale-dependentDFAslopeαonthetimescalenforallhourly,on-and
off-peakelectricitypricesinAlberta(left),Ontario(center),andMid-C(right)markets.Thepresentedstatisticsrevealcomplexcorrelatedstructure
ofpricemovementswithquasi-periodiccomponentsassociatedwithdailyandweeklycycles. Inallpresenteddatasetsthescale-dependentDFA
exponentissignificantlybelowthelevel1.5definingthestateofinformationalefficiency,whichprovidesanopportunityofforecastingtheprices
overwiderangesoftimescales.
TheDFAtechniquewasappliedtothetime-integratedsignal
k
y(k)= (x(t)−hxi), (1)
Xt=1
inwhichhxiistheaveragevalueofthehourlyelectricitypricexandk =1,...,N,whereN isthenumberofpointsin
thestudiedtimeseries. Theintegratedsignaly(k)wasdevidedinto M = N/nnon-overlappingsubintervalsofequal
length n rangingbetween 4 and 720 hours. The boxeswere indexedby m = 1,...,M and their starting times were
labeledbyk . Foreverybox,theleastsquareregressionliney (k)representingthelocallineartrendinthatboxwas
nm nm
fittothedata. Usingthesefits, theintegratedseriesy(k)waslocallydetrendedandtherootmeansquarefluctuation
oftheresultingdetrendedsignalwascalculated. ThedescribedcalculationwasrepeatedforeachoftheMboxesand
theresultingvalueswereaveragedtoobtainthecharacteristicdependenceonthetimescale:
1 M 1knm+n
F(n)= vutM n (y(k)−ynm(k))2 (2)
Xm=1 kX=knm
Forafractal(self-similar)financialtimeseriesx(t),thepower-lawrelationbetweentherootmeansquarefluctua-
tionF andthetimescalenisexpected[44,43]:
F(n)∼nα, (3)
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inwhichαistheDFAscalingexponent[32,33].
Note that our definition of the DFA exponent differs from that used in the majority of other studies of price
fluctuationsoperatingwithlogarithmicpricereturns,inwhichcaseαservesasaproxytotheHurstindexH.Because
thetimeseriesoftheelectricitypricesarequitespiky(especiallyinAlberta),theDFAexponentsevaluatedusingthe
standard return-based approach would be close to zero, making quantitative cross-market comparisons statistically
unreliable. To improve the accuracy of our analysis, we apply the DFA method directly to the time series of the
electricitypricesratherthantothepricereturns,whichisconsistentwiththeoriginalformulationofthemethod[33].
Sinceweomitthestepofcalculatingpricereturns,theresultingDFAexponentisgreaterthanthatderivedfromthe
returnsbyone,andsoα=H+1.
Keepingthisrelationshipinmind,efficientfinancialmarketsdescribedbyrandomBrownianwalkmodels[45,29]
exhibitthevaluesofαwhicharecloseto1.5signalingtheabsenceofcorrelationsbetweenthepriceincrements[28].
Ithasbeenshown [7, 46] that deregulatedelectricity marketsdo notsatisfy this conditionandin that sense are not
efficient. Inaddition,αvaluesofelectricitymarketstendtovarywithscalereflectingcomplexstructureoftheprice
dynamicsinvolvingquasi-periodiccyclesandrandomvariations[7].
Fig. 2showsscale-dependentbehavioroftheDFAfunctionsF(n)characterizingstudiedelectricitymarkets. The
local values of α exponentswere computedwithin narrown intervalsof exponentiallyincreasing width ensuringa
uniformbinningonalogarithmicscale.ItcanbeseenthatCanadianmarketsobeytheanti-persistentconditionα<1.5
acrossalltemporalscales. TheMid-Cmarketshowsapersistentbehaviorcharacterizedbyα > 1.5atn < 12. This
indicatesthatthe daily cycle in this marketdominatesrandomfluctuationsleadingto statistically significanttrends
overtimespansshorterthanthehalf-dayinterval.
Allthreemarketsexhibitanti-persistentregimesatn > 24andoperateinastronglyinefficientstate, confirming
ourpreviousresultobtainedfordailyelectricityprices[7]. Table1providesaverageandminimumDFAindexvalues
forthestudieddata,alongwithspectralanalysisresultsdiscussedbelow.
Itcanbenoticedthatthedropofαassociatedwiththedailyperiodicityisshiftedtowardlargerrelativeton=24.
Thisshiftmaybe caused byan interplaybetweenthe saturationof F(n)and a decreaseof the trendsat time scales
greaterthatthecycleperiod.Qualitativelysimilar,albeitlesspronouncedshiftedsignatureisobservedneartheweekly
periodicity.
TheFourierspectralanalysiswasusedforamoreaccuratedescriptionofperiodiccomponentsofpricedynamics,
and also as an independent test for its informational inefficiency. The power spectrum S(f) is obtained from the
FouriertransformX ofthepricesignalx(t)definedinthecontinuouslimitas
T
T 1
X (f)= x(t)e−i2πftdt, S(f)= X X∗, (4)
T Z0 T T T
whereX∗ isthecomplexconjugateofX andT >>1/f isthelengthofthetimeintervaloftheanalysis. Themethod
T T
was implemented using the standard fast Fourier transform algorithm. For a self-similar signal the spectral power
scaleswiththefrequency f asS(f)∼ f −β wheretheindexβisrelatedtotheDFAindexthroughβ=2α−1,sothat
β=2isthecaseoftheefficientmarket(Fig. 3).
TheobtainedFourierpowerspectraareconsistentwithresultsoftheDFAdiscussedabove(Fig. 3),andreveala
stronglyinefficientbehaviorofthe electricitymarkets. Inaddition,spectraanalysisdemonstratesa relativelystable
dailycycleaswellasalessstablebutevidentweeklyperiodicity.Theaverageβestimatesarestatisticallysignificantly
belowthevalue2. Thevaluesα calculatedbasedonthespectralexponentsareinanapproximateagreementwith
theor
theaverageαvaluesobtaineddirectlyfromtheDFA(thefirstandthelastcolumnsofTable1).
Bothmethodsclearlyshowthattherearestronganti-persistentcorrelationsbetweenthevaluesofelectricityprices
atalltimescales forall-hourtime dataaswellas forthe subsetsoftheon-andoff-peakdata. Thisisevidentfrom
theanalysisofthemaximumDFAindexvalues(secondcolumnitTable1)representingthelargestαacrosstheentire
range of n scales for each market. In most of the data sets, [α(n)] < 1.5 meaning that at no scale the markets
max
becomeefficientorpersistent. TheonlyexceptionfromthistendencyistheMid-Cmarketatn < 12hoursshowing
greaterthan1.5DFAindexvaluesforbothall-hourandon-peakprices. Thiseffectisabsentintheoff-peakMid-C
datasuggestingthattheshort-terminformationalpersistencyofthismarketisafootprintofthehighdemandperiods.
DespitethesignificantvariabilityoftheDFAexponentsacrosstemporalscales, theaverageαvaluesofall-hour
pricesinAlbertaandMid-CmarketsshowninTable1areinanagreementwiththosedescribingdailypricesinthese
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Figure3:FourierpowerspectraS(f)forall-hour,on-andoff-peakelectricitypricesinAlberta(left),Ontario(center),andMid-C(right)markets.
Inalldatasetstheaveragespectralexponentβislowerthan2,whichconfirmstheconclusionoftheDFAanalysisregardingtheinefficientbehavior
ofthestudiedpricedynamics.CanadianmarketsarecharacterizedbysignificantlysmallervaluesofβcomparedtotheAmericanMid-Cpool,and
thereforehaveahigherdegreeofinefficiency.
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Table1:AveragevaluesofDFAandspectralexponentsofhourlyelectricityprices.
Timeseries <α(n)> [α(n)] β α =(β+1)/2
max theor
Alberta,all-hoursprices 0.87±0.13 1.18±0.02 0.93±0.07 0.97±0.54
Alberta,on-peakprices 0.87±0.08 1.09±0.01 0.94±0.06 0.98±0.53
Alberta,off-peakprices 0.83±0.09 1.10±0.04 0.81±0.07 0.91±0.54
Ontario,all-hourprices 0.87±0.13 1.16±0.01 0.92±0.06 0.96±0.53
Ontario,on-peakprices 0.86±0.06 1.03±0.04 0.83±0.06 0.92±0.53
Ontario,off-peakprices 0.91±0.08 1.09±0.05 0.89±0.05 0.95±0.53
Mid-C,all-hourprices 1.17±0.19 1.69±0.04 1.95±0.04 1.48±0.52
Mid-C,on-peakprices 1.21±0.11 1.62±0.02 1.80±0.03 1.40±0.52
Mid-C,off-peakprices 1.22±0.05 1.33±0.04 1.63±0.06 1.32±0.53
marketsaccordingtoourprecedingstudy[7]: < α(n) >= 0.90±0.08forAlberta,< α(n) >= 0.90±0.08forMid-
C. To complementthese earlierreportedfindings, we also conducteda DFA analysisof daily Ontariopriceswhich
yielded< α(n) >= 0.93±0.08. Bycomparingthesenumberswiththe< α(n) >valuesinthetable,onecanseethat
thediscrepancybetweenthetwosetsofDFAexponents(theonesobtainedfordailyandhourlyprices)iswithinthe
statistical uncertainty of our measurements. The fact that the 24-houraggregationof the prices does not affect the
averageDFA exponentindicatesthatthe latter isinsensitiveto the short-scalefluctuationsofthe demanddrivenby
thedailysocioeconomicdynamics,includingthe24-hourcycleandintradaynonstationarities.
Overall, the correlated price movements in all three markets present an opportunity of forecasting their future
dynamics based on the historic behaviors over essentially all time scales involved. The Fourier spectral analysis
confirmsthewellknownexistenceofdailycycleswhichcanbeusedasanauxiliaryfactorinthepriceforecasting.
3.2. Probabilitydistributions
Toverifythestabilityofhighermomentsofelectricityprices,weinvestigatedtheParetoprobabilitydistribution
ofhourly,on-andoff-electricity data(Fig. 4). The Paretoexponentsobtainedfromthis analysisprovidea critical
pieceofinformationonwhetherthestatisticalpredictionofthepricescanyieldrobustresults.
The probability distributions were estimated using the normalized discrete histograms N(x) with constant bin
width chosen to be 5 currencyunits for Alberta and Ontario marketsand 1 unitfor the Mid-Cmarket. The rate of
decay of the histogram tails was described by the Pareto index γ [47] defined by the power-law fit N(x) ∼ x−(γ+1)
appliedtoaselectedpricerange. Asfollowsfromthedefinition,theParetoindexdescribestheasymptoticshapeof
thecomplementarycumulativedistributionfunction.Thevaluesofγwerecomputedusingthestandardmean-square
fittingalgorithmappliedonthelog-logscale.
DependingonthenumericalvaluesoftheParetoindex,thefollowingcasesarepossible: ifγ ≤1,neithertheav-
eragevalueofpricenoritsstandarddeviationcanbedetermined;if1<γ≤2,theaveragepricecanbecalculatedbut
thestandarddeviationcannot;γ >2meansthatbothparameterscanbeevaluatedandtheresultsofpriceforecasting
arepotentiallyrobust. ThevaluesoftheParetoexponentobtainedinourstudy(Fig. 4,Table2)indicatethatallthree
marketspasstherequiredminimumlevelγ=2atleastforthepricerangex∈[1,200].Mid-Celectricitypricesdonot
haveaheavy-taileddistribution,sothereisnoriskofunexpectedlyhighfluctuations,whereastheCanadianmarkets,
especiallyAlberta,showpronouncedfattails. TheprobabilitydistributionsdescribingtheAlbertamarketapparently
hastwocomponents–thePoissonian-likecentralportionwithawell-definedmaximum,andaheavypower-lowtail.
InTable 2, this behavioris demonstratedbydividingthe distributionsintotwo intervals. Pareto indexesof Alberta
electricity prices in the range x ∈ [200,1000]are lower than in the range x ∈ [1,200] and correspondto the case
1 < γ ≤ 2. Althoughthisdoesnotexcludethe possibility of forecastingthe averageprice, such predictionwill be
statisticallyunstable,especiallyforpriceswhicharesignificantlyhigherthantheaverage. Ontarionumbersshowthe
oppositetendency,withtherange x ∈ [200,1000]exhibitingafasterpowerlowdecaycomparetotheotherstudied
price interval. Despite the presence of the fat tail, this market allows for robust statistical prediction even for the
highestvaluesoftheprice.
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Figure4:Normalizedprobabilityhistogramsofall-hour,on-andoff-peakelectricityprices.ThepresentedvaluesoftheParetoexponentγindicate
apossibilityofrobuststatisticalpredictionofmeanpricevaluesforallthreemarketssincetheconditionγ>1ismet.OntarioandMid-Cmarkets
arealsocharacterizedbystablestandarddeviationsduetoγ>2,whiletheAlbertamarketdoesnotsatisfythisrequirement.
Table2:ValuesoftheParetoindexγfortworangesofelectricitypricesinthestudiedmarkets.
Timeseries x∈[1,200] x∈[200,1000]
Alberta,all-hourprices 2.072±0.053 1.318±0.068
Alberta,on-peakprices 2.360±0.084 1.273±0.070
Alberta,off-peakprices 2.394±0.084 1.670±0.082
Ontario,all-hourprices 3.016±0.064 3.508±0.045
Ontario,on-peakprices 3.177±0.067 4.106±0.128
Ontario,off-peakprices 2.964±0.075 3.095±0.115
Mid-C,all-hourprices 4.284±0.190 -
Mid-C,on-peakprices 4.453±0.155 -
Mid-C,off-peakprices 4.310±0.190 -
8
AllthreemarketsseemtobeinsensitivetothedemandlevelshowingcomparableParetoexponentsforall-hour,
on-andoff-peakdatasetsforbothpriceranges,confirmingourearlierobservations[7]. Itremainstobeunderstood
whythemarketdemandmakesnonoticeablecontributiontothestatisticalpropertiesoftheprocess.
4. Forecastingelectricitypricemovements
Since anti-persistency implies a stochastic process with negative correlation between its increments, our next
step is to investigate statistical interdependenceof previousand currentprice movements. To achieve this goal we
introduceda new methodofmultiscale incrementssensitiveto such correlations. The multiscale increments∆x(n)
i
onthescalenarecomputedaccordingto
1 ti−1 ti+n
∆x = x(t)− x(t) (5)
i n 
t=tXi−n−1 Xt=ti 
andrepresentarunningdifferenceofsubsequentaggregatedpricevaluesobtainedbyaveragingx(t)innon-overlapping
binsof width n. The statistical relationship ∆x versus∆x between the currentand precedingincrementscarries
i i−1
informationaboutcorrelatedpricedynamicsatagivenscaleandforagivenpricerange. Ifonaverage∆x ∼ ∆x ,
i i−1
thepricemovementsarepositivelycorrelatedandpersistenttrendsarepresent. Ontheotherhand,if∆x ∼ −∆x ,
i i−1
thecorrelationisnegativerevealinganti-persistentpricebehavior[29,7].
Fig. 5showsresultsoftheincrementanalysisofthethreeelectricitymarketsforafixedtimescalen=1. Thetop
rowofpanelsdisplaysscatterplotsofsuccessivehourlypriceincrementswhichincludebothon-andoffpeakprices.
The characteristic amplitude of price changes is considerably different for the studied markets, with the increment
rangeinMid-Cmarketbeingabout10timessmallerthanthatinAberta.
TheincrementplotsoftheCanadianmarketsexhibitacommonpattern. Datapointsinthefourthquadrantofthe
coordinateplane definedby the conditions∆x < 0 and ∆x > 0 are aligned along a diagonalline with slope −1
i i−1
suggestingstronglyanti-correlatedbehaviorofpriceincrements.
Mid-Cpricesalsoshowacorrelatedregressionpatternbutwithapositivesloperevealingpositivecorrelationsat
thestudiedtimescale. DirectcomparisonofthisMid-CsignaturewithAlbertaandOntariopricesisnotjustifieddue
toadrasticallysmallerrangeofpricefluctuationsintheMid-Cmarket. Becauseofitscompactshape,theregression
plotofMid-CelectricitypricesfitswithinthecoreregionofAlbertaandOntariomarkets.
Besidesthemaintendencyassociatedwiththeanti-correlation,Canadianmarketsshowtwoadditionalfeatures:
alignment of the increment values along the positive vertical axis (∆x > 0, ∆x ≈ 0) and along the negative
i i−1
horizontal axis (∆x ≈ 0, ∆x < 0). The first of these features reflects the tendency of the electricity price to
i i−1
growabruptlystartingfromasteadyconditioncharacterizedbylowpricevariability. Thesecondfeaturereflectsthe
oppositetendencyinwhichthepricedropsafteraprecedingpeakandreturnstoasteadycondition.
The second and third rows of panels in Fig. 5 show scatterplots of price increments constructed for on-peak
and off-peakprices. It can be seen that on- and off-peak prices in Alberta and Ontario demonstrate the same anti-
correlation pattern as the one identified for the whole set of hourly prices in these markets. This pattern is more
pronouncedin the on-peak price movementswhich is an expected result since the amplitude of the price jumps is
muchhigherduringthepeaktimes.
Surprisingly,neitheron-peaknoroff-peakpricemovementsintheMid-Cmarketshowpositivecorrelationsseen
intheoriginalall-hourdata. Apossibleexplanationofthiseffectisthatthecorrelationspresentintheall-hourdata
areproducedbytransitionsbetweenhigh-andlow-demandintervalsoftheMid-Cmarketratherthanbyitsdynamics
duringtheseintervals.
ThelastrowinFig. 5showsaggregatedscatterplotsconstructedbyaveraging∆x priceincrementsoverasetof
i
uniform∆x bins. Theseare constructedtorevealthe prevailingstatistical tendenciescontrollingthepricemove-
i−1
mentsinthestudieddata,enablingtheirvisualandquantitativecomparisons.
It can be seen that the Canadian markets are dominated by the anti-correlated price movements manifested in
thefourthquadrantoftheaggregatedscatterplots. Theplotsexhibitanearlylineardecay,withtheslopecoefficient
of about −0.5 for Alberta and −1 for Ontario market. This tendency is observed across the entire range of price
incrementsforbothon-oroff-peakdatasets. Thelineardependenceofsmallpricemovementswasobstructedinthe
originalscatterplotsbutisevidentintheaggregatedplots.
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Figure 5: Regression diagrams showing current (∆xi) versus preceding (∆xi−1) hourly price increments for all-hour, on- and off- peak data.
RegressionplotsforAlbertaandOntariomarketsaresimilarinshapeandrevealnegativecorrelationsofpricemovementsassociatedwiththespiky
structureofthedata. TheMid-Cpriceincrementsexhibitpositivecorrelationsinall-hourdataonly. Bottompanelsdisplayaveragedversionsof
thediagrams,with∆xivaluesaveragedover∆xi−1binsofthesamewidth.
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