Analysis of relationships between hourly electricityprice and load in deregulated real-time powermarkets

K.L. Lo and Y.K. Wu

Abstract: Risk management in the electric power industry involves measuring the risk for allinstruments owned by a company. The value of many of these instruments depends directly onelectricity prices. In theory, the wholesale price in a real-time market should reflect the short-runmarginal cost. However, most markets are not perfectly competitive, therefore by understandingthe degree of correlation between price and physical drivers, electric traders and consumers canmanage their risk more effectively and efficiently. Market data from two power-pool architectures,both pre-2003 ISO-NE and Australia’s NEM, have been studied. The dynamic character ofelectricity price is mean-reverting, and consists of intra-day and weekly variations, seasonalfluctuations, and instant jumps. Parts of them are affected by load demands. Hourly signals onboth price and load are divided into deterministic and random components with a discrete fouriertransform algorithm. Next, the real-time price–load relationship for periodic and random signals isexamined. In addition, time-varying volatility models are constructed on random price and randomload with the GARCH (1,1) model, and the correlation between them analysed. Volatility plays acritical role on evaluating option pricing and risk management.

1 Introduction

Ideally a competitive market should give the correct short-term and long-term price signals to buyers and sellers. In theshort term, prices should reflect the short-run marginal costof production. In other words, energy buyers should nothave to pay more than what it costs to produce and deliver.It should also give incentives for investment in generationand transmission capacity in the long term. The ideal resultis price stability. However, in the single-settlement real-timepower market, electricity prices exhibit high volatility andinclude a number of price spikes. The price spikes in the USMidwest in June 1998 were a striking example. Electricityprices rose to $7,500 per megawatt hour compared withtypical prices of around $30 per megawatt hour. Nowadays,it is not surprising that power prices are the most volatile ofall traded commodities.

Electricity prices exhibit short-term high volatility due tophysical market characteristics. Therefore an importantaspect of assessing the markets is to understand thebehaviour of electricity prices, and the relationship betweenphysical market characteristics and electricity price. Themain objective for exploring the relationship between priceand physical factors is to accurately predict price and itsvolatility. In general, there are two types of electricity priceforecasting methods: one is the mathematical modellingmethod such as time-series analysis and ANN; the other isthe physical simulation module, which includes a detailed

model of the power system and a procedure for pricing.Although the simulation method can provide detailedinsights into the price curve, it requires many systemparameters and a lot of market information. For example,the ISO-NE system consists of 324 generating units in 180market nodes. There are about 2500 constraints and 3800variables [1]. Therefore it is difficult to implement priceforecasting using the physical module in a large powersystem because of the number of system uncertainties. Asfor the mathematical modelling method, an importantprocedure is to find the inclusion of important exogenousvariables that would impact the price forecasting results.Some important physical variables [2–4] can be summed as:

� Power demands

� System or regional reserves

� Transmission and generation outages

� Bidding strategy and market power

� System congestion

� Ancillary services

� Market rules

Among these factors, most are unpredictable except forpower demands. Moreover, power demands play the mostimportant role in the behaviour of periodic electricity price.These are the reasons why this paper focuses on the price–load relationship; the results could enhance price forecastingby means of accurate demand forecasting. It is apparentthat other physical variables could also impact electricityprices, especially on the random components of prices.However, it is not easy to consider the effect of thesevariables on prices because little public information isavailable from the pre-2003 ISO-New England andAustralia’s National Electricity Market (NEM).

The authors are with the Power Systems Research Group, Department ofElectronic and Electrical Engineering, University of Strathclyde, Glasgow G11XW, UK

r IEE, 2004

IEE Proceedings online no. 20040613

doi:10.1049/ip-gtd:20040613

Paper first received 31st March 2003 and in revised form 28th January 2004

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 441

On the issue of price–load relationship, [5] introduces therelationships in California’s electricity market with regimeclassification and regression analysis. Reference [6] presentsa fuzzy-set based model to illustrate the price–load relation.However, these proposed methods consider the relationshipas a pure regression problem. In fact, some price signalscould not be described as price–load regression functionsbecause they may be affected by other physical factors.Therefore in this paper, the price–load relation is dividedinto two parts: periodic and non periodic components thatare used to evaluate reasonably the effect on real-time pricesdue to load fluctuations. These periodic components arefiltered by a typical frequency-based method, the DFTalgorithm. On the other hand it is quite difficult to evaluateany clear connection between random price and randomload. However, it could be useful to study the probabilitydistribution of correlation coefficients to estimate the price–load relationship according to different periods and loadlevels. In addition, the high volatility on electricity prices is apotential problem in deregulated power markets. Volatilityis the most important variable on option pricing, portfoliooptimisation and risk management. Therefore the volatilityof random prices and loads as well as the correlationbetween them has been analysed in this paper.

2 Characteristics of competitive electricity prices

Power price volatility or spikes cannot be avoided even inperfect markets. It is important to understand thecharacteristics of competitive electricity prices from histor-ical data in different power markets. For example, pricevolatility in the electricity market is rooted in hourly, daily,weekly, and seasonal uncertainty; most of the factorscontributing to the uncertainty are reflections of regularfluctuation of the power loads. The periodic fluctuationcould be considered as the deterministic component ofelectricity prices. Furthermore, electricity prices have anumber of instant price spikes with a long-term mean-reverting trend, which is the random component and resultsfrom the physical characteristics of the power system, suchas unusual demand, unexpected generator outages andtransmission constraints. Factors affecting the fluctuation ofrandom price result from not only load conditions, but alsosupply conditions.

Figure 1 shows the static relationship between supply-load sides. The aggregate supply curve is the ranking of allparticipative generation units, and the real-time price couldbe determined by the intersection between supply and loadcurves. If the load curve is given by D1, the price is notsensitive to load shift. On the other hand., if the load curveis given by D2, even small increments in load would havehuge effects on the price because the load intersects with thesteep part of the supply stack. In addition, the supply and

load curves are not static in time. Therefore the relationshipbetween load and prices in real-time power markets hasbecome an important topic in power system economics.

For the analysis of deterministic components it isessential to determine the linkage of a time-based orfrequency-based function between price and load. Anysignal can have more than one representation to identify itdepending on the ability of the chosen representation toextract important information from the signal. A signalmay appear very complex in the time domain, but may havea simple representation in the frequency domain, and viceversa. For example, most financial data for stock marketsare usually processed with time-series analyses because thesesignals show time trends. As for frequency-based analysis, itprovides a seasonal effect or a cyclic view on the data in thefrequency domain. Because the typical curves of electricityprices and loads have inherent periodic components it isuseful to obtain their deterministic components underfrequency-based analysis. On the other hand, for therandom components of prices and loads, it is difficult todecide the deterministic function between them. However,the probability that electricity price fluctuation results fromunusual load fluctuations could be evaluated with thestatistical analysis of historical data.

In addition, price spikes could lead to high pricevolatility. These price spikes do not follow a regular dailypattern; they appear to be an intrinsic problem with aparticular competitive market [7]. If the random manner ofvolatility on price and demand can be detected in advance,the trader in the power market could predict the risk bysimulation methodology [8] on the basis of time-varyingvolatility, and the effect of random load volatility on theprice volatility could also be evaluated.

3 Market designs on pre-2003 ISO-NE andAustralia’s NEM

In this work two power markets, ISO-NE and Australia’sNEM, were chosen for analysis. The pre-2003 ISO-NEmarket was an hourly real-time wholesale market, whichfollows a single energy clearing price system. NEM inAustralia is a half-hourly real-time wholesale market. Itspricing system is based on the regional reference price(RRP) where a uniform price is computed for each region.The main similarity between both markets is that they aredesigned as single-settlement real-time markets. In compar-ison with multisettlement systems, price fluctuations insingle-settlement systems are highly unstable because theyare affected by all system uncertainties only during actualoperation periods. Risk management needs to pay moreattention to the electricity price characteristics in thesemarkets. This is the reason why these two markets arechosen for analysis.

The main differences between pre-2003 ISO-NE andNEM were the trading intervals and pricing models. Theuniform-pricing (UP) market model with one hour tradinginterval was adopted in the pre-2003 ISO-NE. The marketarchitecture in NEM is the regional-pricing (RP) marketmodel with an half-hour trading interval. The half-hourtrading interval provides more market information toparticipants, which would improve the accuracy of demandor price forecasting. This is shown, in Table 1 in Section 4,where the number of bins in NEM is higher than that inISO-NE. In addition, the development of the RP-basedmarket is an improvement over the UP-based marketbecause RP markets could reflect the actual balancecondition between supply and demand in each regionto give the correct price signals. This leads to a higher

aggregate supply curve

D1

D2

load curve

real

-tim

e el

ectr

icity

pric

e

power, MW

Fig. 1 Real-time equilibrium dynamics between supply and loadsides

442 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

price–load relation because the effect of inter regioncongestion on price would be reduced. Therefore the resultsin Section 4 indicate that the percentage of periodiccomponents in the NEM price signal is higher than thatin the ISO-NE price signal. In this analysis the regionalprices and demands in the New South Wales (NSW) regionof NEM were adopted.

To avoid price risk exposure, the market architecturein ISO-NE has moved from a single-settlement system toa multisettlement system since 1st March 2003. Themultisettlement design consists of day-ahead and real-timemarkets, which allows market participants to secureday-ahead prices in the financial market and reducetheir vulnerability to price fluctuations in the real-timemarket. In Australia’s NEM, there is also a separatefutures market that is traded via the Sydney FuturesExchange and the Australian Stock Exchange. Thishedging market trades futures contracts and runs in parallelwith the real-time power market. Therefore identifyingand understanding the price–load relationship in a real-timemarket would enable market participants to makemore informed decisions about their bids and offers infinancial hedging markets, such as day-ahead, hour-ahead,futures, or forward markets.

4 Periodic components in electricity price andpower demand

It is clear that the regular component of price dynamicsis derived from load-driven effects. In this Section theperiodic components in both price and load are obtainedbased on the discrete Fourier transform (DFT) algorithm[9, 10]. The Fourier transformation decomposes a waveforminto sinusoids of different frequencies that sum to theoriginal waveform. It is a useful tool to transfer priceand load signals from a time base to a frequency base.The discrete Fourier transform (DFT) of a discrete signal

x(n) is defined as

X kð Þ ¼PNn¼1

x nð Þ exp �j n�1ð Þ k�1ð Þ2pN

� �k ¼ 1; :::;N ð1Þ

where N is the number of data sample, j ¼ffiffiffiffiffiffiffi�1p

, and theinteger k is defined as the bin number which gives the DFTvalue X(k) corresponding to a specific frequency fk. Forexample, in the ISO-NEmarket, the data is sampled at onceper hour, and the DFT has a length of 600 sample data,which means that each bin is 0.001667(1/h) wide and the binnumber 26 is centred on the frequency of 0.041667 (1/h).The mathematical functions among these values are givenby (2) and (3)

binwidth 1=hð Þ¼sample rate ð1=hÞ =DFT length ð2Þ

bin frequency ð1=hÞ ¼ bin number� 1ð Þ�binwidth 1=hð Þð3Þ

On the other hand, the inverse DFT (IDFT) that relatessamples from the frequency domain to the time domain isgiven by

x nð Þ ¼ 1

N

XN

k¼1X kð Þ exp j n� 1ð Þ k � 1ð Þ2p

N

� �n ¼ 1; :::N

ð4Þ

The processing operation of the IDTF is similar to that ofthe DFT except that the phase of the exponential isreversed. Figure 2 shows the frequency-domain character-istic of energy clearing prices (ECP) and system load forISO-NE in December 2001; the horizontal line representsperiod and the vertical line is called the power spectrum ofthe signal, which represents power as a signal. Similarly thepower spectrum of regional electricity price and load inNew South Wales, Australia is shown in Fig. 3.

Table 1: Statistical results of same bin numbers between price and load spectrum for both power markets

Power market Duration Identical bin numbers and corresponding periods (inparenthesis) in spectrum between prices and loads

ISO-NE 4/2B5/6 2001 weekday 1, 26(24h), 51(12h), 126(4.8h)

weekend 1, 11(24h), 21(12h), 41(6h)

6/4B7/8 2001 weekday 1, 26(24h), 51(12h), 126(4.8h)

weekend 1, 11(24h)

8/13B9/16 2001 weekday 1, 26(24h), 51(12h), 101(6h)

weekend 1, 11(24h), 31(8h)

10/1B11/4 2001 weekday 1, 26(24h), 51(12h), 101(6h)

weekend 1, 21(12h), 31(8h)

11/5B12/9 2001 weekday 1, 26(24h), 51(12h),76(8h),101(6h), 126(4.8h), 176(3.43h)

weekend 1, 11(24h), 31(8h), 61(4h)

NEM-NSW 4/2B5/6 2001 weekday 1, 26(24h), 51(12h), 76(8h), 101(6h)

weekend 1, 11(24h), 21(12h), 31(8h), 41(6h), 51(4.8h)

6/4B7/8 2001 weekday 1, 26(24h), 51(12h), 76(8h), 101(6h)

weekend 1, 11(24h), 21(12h), 31(8h), 51(4.8h), 61(4h), 81(3h)

7/9B8/12 2001 weekday 1, 26(24h), 51(12h), 76(8h), 101(6h), 126(4.8h)

weekend 1, 11(24h), 21(12h), 31(8h), 41(6h), 51(4.8h), 81(3h)

10/1B11/4 2001 weekday 1, 26(24h), 51(12h), 76(8h), 101(6h)

weekend 1, 11(24h), 21(12h), 31(8h), 51(4.8h)

11/5B12/9 2001 weekday 1, 26(24h), 51(12h), 76(8h), 101(6h)

weekend 1, 11(24h), 21(12h), 31(8h), 41(6h)

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 443

It is clear from Figs. 2 and 3 that the time-seriesof electricity prices or load can be transferred successfullyinto some specified frequencies in the frequency domainwith the DFT algorithm. For example, the major periodiccomponents identified from Fig. 2a correspond toperiods of 24 , 12, and 8 h. Comparing (a) and (b) in bothFigs. 2 and 3, one observes that the periodic signal inreal-time electricity price is more complicated and muchvaguer than that with the real-time electricity loadbecause the time-series of real-time price is very volatileand consists of a number of instant spikes. However, themain periodic components in both price and demand arealmost identical.

In this work, the same frequency components betweenprices and loads are identified according to their spectrawith bin numbers. For example, the spectra for weekdayhourly prices and loads are shown in Figs. 4a and 4b,respectively. The observed data derived from the ISO-NEinvolves successive five weekday data (600h). In Fig. 4 thefirst bin number in both power spectrums is called theDC component of the signal, which represents the sumof all discrete signals. The value of the DC componentis proportional to the average value of the originalsignals. Other bin numbers, such as the 26th, 51st, 76thand 126th, are the fundamental components of thesignal. These bin numbers are associated with specificperiod components. For example, the 26th bin number

3000

2500

2000

1500

1000

500

00 10 20 30 40 50 60 70 80

pow

erpo

wer

, × 1

06

period, h/cycle

a

0 10 20 30 40 50 60 70 80period, h/cycle

b

1.4

1.2

1.0

0.8

0.4

0.6

0.2

0

Fig. 2 Power spectra for electricity price and load signals in ISO-NE marketa Electricity priceb Electricity load

3000

2500

2000

1500

1000

500

00 20 40 60 80 100 120 140 160

pow

erpo

wer

, × 1

06

period, ½h /cycle

a

0 20 40 60 80 100 120 140 160

period, ½h /cycle

b

1.4

1.2

1.0

0.8

0.4

0.6

0.2

0

Fig. 3 Power spectra for electricity price and load signals inNEM-NSW marketa Electricity priceb Electricity load

1st

26th

51st

76th126th 476th

526th

551st

576th

1st

26th

51st

101st126th

476th501st

551st

576th

6000

8000

10000

4000

2000

00 100 200 300 400 500 600

pow

erpo

wer

, × 1

06

1.6

2.0

1.2

0.8

0.4

0

bin number

a

0 100 200 300 400 500 600

bin number

b

Fig. 4 Spectral content of price and load variation with binnumbersa Electricity priceb Electricity load

444 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

(k¼ 26 in (1)) represents the period of one day (24 h)and the 76th bin number (k¼ 76 in (1)) indicates theperiod of eight-hour variation. From Fig. 4 it is evidentthat there are similar periodic components appearing in theprice and load spectra. In this sample data the sameperiodic components involve the bin numbers of 26th,51st, 126th, 476th, 551st and 576th. There is the symmetryproperty that applies to the DFT of a real signalsequence. The bin numbers of the DFT components from1 to N/2+1 thus repeats from N/2+1 up to N in a mirror-image fashion. This is proved in the Appendix (Section 9).That is, the magnitude of bin 26 is identical to that of bin576, bin 51 and bin 551 are the same, and bin 126 andbin 476 are the same. Therefore above the bin number ofN/2+1 the DFT results would be redundant, andthus these redundant bin numbers could be eliminatedfrom the spectra.

After extracting the periodic signals in the frequencydomain these same periodic signals in price and load couldbe restored to the time domain by inverse Fouriertransformation based on (4). The results of restoring theperiodic and the random components of the price and loadsignals to the time domain are illustrated in Figs. 5a to 5d,respectively. In Figs. 5a and 5c the solid lines represent theoriginal curves and the dashed lines represent the mainperiodic curves with the same frequency componentsbetween prices and loads, respectively. Figures 5b and 5dshow, respectively, the random prices and random loadsderived from the residuals after filtering out the periodiccomponents.

In addition to the above sample, other data ofprices and loads in both power markets have been used inthis paper to analyse the identical periodic components.A total of ten-month price–load periodic relations inISO-NE and NEM-NSW are examined. The same periodicbin numbers (and corresponding period values) betweenloads and prices for each month are summarised inTable 1. The bin numbers in the spectra indicate thatsome specific frequencies exist between prices and demandsin both power markets. These periodic variations of pricesare derived from the variations of load with specificfrequency components. In this analysis the percentages ofaverage periodic components filtered by the DFT are 79.23and 94.53% for the ISO-NE whole prices and loads,respectively, and 82.36 and 96.77% for the NEM-NSWwhole prices and loads, respectively. Weekdays andweekends were considered separately because the ampli-tudes on loads and prices on weekends were lower thanthose on weekdays.

Based on the identical bin numbers from historicaldata, the effect of load variation on the periodiccomponents of price appears to be predictable in advance.For example, the fundamental spectrum components ofweekday price signals in ISO-NE include the binnumbers of 26 and 51, and in NEM-NSW market arethose of 26, 51, 76, and 101. The higher bins indicatethat the effect of loads on the periodic components ofthe price is high. That is, the NEM-NSW represented ahigher price–load periodic relation than that in theISO-NE because it implements better trading-interval andpricing models. In addition, the effect of load on price atweekends is lower than that in weekdays for the ISO-NEbecause the price signals were very fluctuant duringweekends. Removing these periodic signals, one coulddetermine the characteristics of the random componentin price and demand. Moreover, it is more objective touse the random component to estimate the volatility ofelectricity price.

140

120

100

80

60

40

20

0

−20

100

80

60

40

20

0

−20

−40

0 100 200 300 400 500 600

time, h

b

a

d

c

syst

em p

rice

syst

em p

rice

with

rem

ovin

g pe

riodi

c co

mpo

nent

ssy

stem

dem

and

with

rem

ovin

g pe

rodi

c co

mpo

nent

s

1.0

1.7

1.5

1.3

1.1

0.9

syst

em d

eman

d, M

W ×

104

4000

3000

2000

1000

−1000

−2000

0

Fig. 5 Time-domain signals for periodic and random componentsof price and loada Electricity price and its periodic componentb Random price after removing periodic componentc Electricity load and its periodic componentd Random load after removing periodic component

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 445

5 Random components in electricity price andload

In the previous Section identical periodic components areidentified and removed by DFT and IDFT. In this Sectionthe weekday random components after filtering out periodiccurves are analysed using correlation analysis. Theseanalyses of random components are based on the wholedata sample in 2001. In addition, the probability distribu-tions of random electricity prices and random loads areillustrated in this Section. Probability distribution can offertraders a better idea of the ranges of variable movement as aresult of different periods and loads. For example, Table 2shows the characteristics of random prices and loads byusing the probability distribution of variables on peak hours(from 8:00 to 20:00) and off-peak hours (from 20:00 to 8:00).The value of skewness measures the coefficient of asymmetryof a distribution. If skewness is negative, the data are spreadout more to the left of the mean than to the right. The valueof kurtosis is a measure of whether the data are peaked orflat relative to a normal distribution. Data sets with highkurtosis tend to have a distinct peak near the mean and haveheavy tails. For a normal distribution the skewness andkurtosis are, respectively, 0 and 3. Further, the Jarque–Berastatistic (JB) test is utilised to evaluate if the valuable has anormal distribution. A distribution is normal when the valueof JB is equal to 0. The JB value is defined as:

JB ¼ n6

S2 þ 1

4K � 3ð Þ2

� �ð5Þ

where

S ¼

Pni¼1

Ri � �RRð Þ3

ns3ð6Þ

K ¼

Pni¼1

Ri � �RRð Þ4

ns4ð7Þ

where n represents the sample size, S and K represent theskewness and kurtosis, respectively, R and �RR indicate thepresented variable and its average, and s represents thestandard deviation of R.

Based on the results in Table 2, the probability distribu-tion of random electricity price on peak hours is skewedleftward with a median well below the mean, and a veryhigh kurtosis statistic. This is typical of random electricity

prices during peak hours. However, these characteristics donot exist in the random loads and the random price duringoff-peak hours.

Furthermore, the correlation between random prices andrandom loads has been analysed in this work. Statisticalresults for ISO-NE 125 peak and 125 off-peak periods in2001 whole sample data are presented, respectively, inFigs. 6a and 6b, which are the probability distributions ofdaily correlation coefficients between loads and prices.

Table 2: Comparison of random prices and random loads based on peak and off-peak probability distribution for wholesamples in 2001

Powermarket

Item Duration Mean $,MW

Median $,MW

Max $,MW

Min $,MW

Std. dev. Skewness Kurtosis Jarque–Bera

price peak �0.06 �1.76 90.51 �40.5 12.28 2.31 13.91 1

ISO-NE off-peak 0.25 0.93 87.84 �32.3 8.76 0.10 10.97 1

load peak �1.76 8.62 4363 �5099 1312 �0.19 5.19 1

off-peak 23.92 �7.30 3989 �3856 889.8 0.13 5.16 1

Powermarket

item duration Mean A$,MW

MedianA$, MW

Max A$,MW

Min A$,MW

Std. dev.A$, MW

Skewness Kurtosis Jarque–Bera

NEM- price peak �0.02 �1.04 84.85 � 17.6 8.21 2.96 23.12 1

NSW off-peak �0.22 0.01 31.32 �17.6 5.20 �0.17 4.59 1

load peak 6.94 31.46 1629 �2702 444.3 �1.22 8.31 1

off-peak �10.8 23.93 1402 �1927 354.6 �0.68 6.23 1

1.0

0.8

0.6

0.4

0.2

00−1.0 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1.0

prob

abili

ty1.0

0.8

0.6

0.4

0.2

0

prob

abili

ty

correlation coefficient

a

0−1.0 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1.0

correlation coefficientb

cumulative

cumulative

Fig. 6 Probability distribution for daily correlation coefficients forISO-NE whole sample dataa Peak hoursb Off-peak hours

446 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

Comparing Figs. 6a and 6b, notice that higher correlationbetween random loads and random prices exists in peak-hour period. Table 3 gives a summary for the correlationanalysis in both power markets for the whole data sample in2001. It also indicates the same conclusion that the price–load correlation is higher in peak hours than in off-peakhours.

In addition to the time factor, the amplitude of loads alsoplays an important role affecting electricity prices. Forexample, price spikes generally occur during peak loads. Toillustrate the distribution of electricity price in different loadlevels, we utilised the cum

ffiffiffiffif3p

rule to separate randomloads into the appropriate three groups of loads namelylow, medium and high strata. The cum

ffiffiffiffif3p

rule is analgorithm to determine the suitable stratum boundaries,and the detail about its theory is illustrated in [11, 12]. It isin fact the cumulative sum of the cube root of the frequencyf of load appearing in a given interval. Based on the cumffiffiffiffi

f3p

rule, the first step is to arrange the load variable into anascending order, and the frequency of certain load appearedin each equal interval is measured. The second step is tocalculate the cube root of each frequency and to computethe cumulative sum of them progressively. Finally, thethresholds of each stratum are constructed on the basis offorming equal intervals on the cum

ffiffiffiffif3p

scale.

In our work the random loads in ISO-NE consist of 3000weekday hourly load data. Following the cum

ffiffiffiffif3p

rule andthree layers of strata setting, an equal interval on the cumffiffiffiffi

f3p

scale could be calculated as 85.2089/3¼ 28.403(Table 4), which yields the two threshold regionsr1¼�13142.2MW and r2¼ 578.1MW. Figure 7 showsthe presented random load series and selected thresholdregions by the cum

ffiffiffiffif3p

rule. Therefore the random load isdivided into three strata: low stratum (�5098.7B�1314.2MW), medium stratum (�1314.2B578.1MW),and high stratum (578.1B4362.6MW). In addition thesame procedure using cum

ffiffiffiffif3p

rule has also been applied tothe random loads in NEM-NSW, and the threshold regionsare obtained as r1¼�428.4MW and r2¼ 113.MW.

Under each stratum in the random load series, thedistribution of corresponding electricity prices has beenanalysed. The statistical result is shown in Table 5 based onthe whole data sample in 2001. According to the statistics inthis Table 6 the skewness value of random electricity pricesin high-load strata is high, which represents some pricespikes existing in the high-load strata. However, there arealso high random prices appearing at the low-load strata.

For example, the peak price in the low-load stratum ofNEM-NSW reaches AUD $62.22 perMW. This price spikecannot be explained with load alone; other factors shouldbe considered.

High random prices would result in high financial risk.To quantify the risk in each load stratum of one could usevalue at risk (VAR) as a measurement tool. VAR is a well-knownmethod of risk measurement [13]. Its definition is thelargest likely loss from market risk that an asset will sufferover a time interval and with a degree of certainty selectedby the decision maker. In this paper the random price maybe regarded as an uncertainty variable because thedeterministic component of prices has been removed fromthe original price data. In other words, the random price isso difficult to predict that it can be regarded as the likelyloss. Therefore these historical random prices in each loadstratum have been directly used to evaluate the VAR with

Table 3 Correlation analyses between random prices andrandom loads for different periods for whole sample data in2001

Power market Time section Probability, %(correlationcoefficient40.6)

Probability, %(correlationcoefficient40.8)

ISONE peak hourperiods

43.2 15.2

off-peak hourperiods

24.2 5

NEMNSW peak hourperiods

56.8 22.4

off-peak hourperiods

36.7 4.2

Table 4 Stratification by cumffiffiffif3p

rule applied to randomload series

Demand intervals (MW) Frequency (f)ffiffiffif3p

cumffiffiffif3p

�5098.7B�4625.6 6 1.8171 1.8171

�4625.6B�4152.6 10 2.1544 3.9716

�4152.6B�3679.5 12 2.2894 6.2610

�3679.5B�3206.4 10 2.1544 8.4154

�3206.4B�2733.3 15 2.4662 10.8816

�2733.3B�2260.3 38 3.3620 14.2436

�2260.3B�1787.2 73 4.1793 18.4229

�1787.2B�1314.2 100 4.6416 23.0645

�1314.2B�841.10 225 6.0822 29.1467

�841.10B�368.10 389 7.2999 36.4466

�368.10B105.00 839 9.4316 45.8783

105.0B578.1 593 8.4014 54.2797

578.10B1051.1 310 6.7679 61.0476

1051.1B1524.2 171 5.5505 66.5981

1524.2B1997.3 73 4.1793 70.7774

1997.3B2470.3 58 3.8709 74.6483

2470.3B2943.4 30 3.1072 77.7555

2943.4B3416.5 22 2.8020 80.5576

3416.5B3889.6 17 2.5713 83.1288

3889.6B4362.6 9 2.0801 85.2089

6000

4000

2000

0

−2000

−4000

−60000 500 1000 1500 2000 30002500

time, h

load

, MW

threshold 2(r2 = 578.1)

threshold 1(r1 = −1314.2)

Fig. 7 Selected threshold regions by cumffiffiffiffif3p

rule applied torandom load series

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 447

different confidence levels. The confidence level determinedby the decision maker is the probability value 1�aassociated with a confidence interval a. A confidenceinterval gives an estimated range of values that is likely toinclude an unknown population parameter.

Figure 8 shows the probability distribution of randomprices in high-load stratum in ISO-NE, and the associatedVAR can be measured based on different confidence levels.Table 6 gives the values of VAR in different load strata with99 and 95% confidence levels for both power markets,respectively. In comparing the statistical VARs in Table 6note that VAR is almost proportional to load stratum.

The statistical method is a direct way to analyse therelationship between price and load. It is easy to implementbecause only simple methods are needed. Moreover, themethod is very flexible. It could be based on different loadlevels, time periods, or price levels. ANN is anothersuggested technique to construct the filtered price–load

relationship. The ANN method can establish the nonlinearrelationship between price and other physical factors if thereare sufficient historical data. However, compared with thestatistical method, the ANN method may be morecomplicated in implementation.

In addition, a static spectrum cannot fully describe theperiodic components of prices and loads because thespectrum content may change in time. This is the reasonwhy there are still some similar partial trends between thefiltered components. For example, the filtered price signalfollows the trend of the filtered load signal from the 504th tothe 576th hour in Figs. 5b and 5d, respectively. Time–frequency analysis [14] could become a better representationmethod to determine the frequencies and their relativeintensities as time progresses.

6 Volatility in electricity price and load

Volatility is a new estimation technique for price variation[15]. It measures the size of fluctuations in time-series.Traditionally the constant standard deviation is utilised asthe tool for evaluating price variation. However, thevariation of time-series such as price and load signalsappears to change over time. As an illustration, Fig. 9shows the changes of random price during four successivedays from 2nd to 5th April in ISO-NE. From this graphobserve that the change in price is quite stable from the 20thto the 40th hour period because it oscillates within a verysmall range. However, the change in price fluctuates anddisplays more volatility during other hours. Therefore theability to model time-varying volatility is very significant.Moreover, a model of stochastic volatility is also usefulwhen estimating a fair price for a given financial option orother volatility-dependent derivative [16].

Table 5: Comparison of characteristics of random prices in different load strata for whole samples in 2001

Power market Load stratum Mean ($) Median ($) Max ($) Min ($) Std dev. Skewness Kurtosis Jarque–Bera

ISO-NE low �10.26 �9.45 25.12 �40.48 7.44 �0.37 6.34 1

medium �1.27 �1.01 87.84 �27.81 8.45 0.95 13.74 1

high 7.70 5.31 90.51 �19.68 12.58 2.76 14.61 1

Power market demand stratum Mean (A$) Median (A$) Max (A$) Min (A$) Std. dev. Skewness Kurtosis Jarque–Bera

NEM- NSW low �3.38 �3.47 62.22 �15.55 6.28 4.60 44.90 1

medium �1.09 �1.08 60.86 �17.58 5.31 1.72 18.22 1

high 2.20 1.26 84.85 �12.69 7.27 3.78 32.69 1

Table 6: VAR estimation for three demand strata on thebasis of different confidence levels for whole samples in2001

Power market Load stratum VAR 95% 99%

ISO-NE low $0.24 $9.80

medium $11.03 $19.85

high $27.04 $68.25

NEM-NSW low A$4.24 A$14.65

medium A$7.02 A$12.53

high A$12.08 A$30.30

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0 20 40 60 80 100

prob

abili

ty

−20

VAR = 18.5190% cofidence level

VAR = 27.0490% cofidence level

VAR = 68.2599% cofidence level

electricity price, S/MW

Fig. 8 Probability distribution of random prices in high-loadstratum and corresponding VAR based on different confidence levelsin ISO-NE

20

15

10

−5

−10

−15

−20

5

0

0

pric

e

20 40 60 80 100

time, h

Fig. 9 Random electricity price during four successive days (2ndto 5th April) in ISO-NE

448 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

In a financial market, instead of the direct financial time-series, volatility measurement is usually calculated from therate of financial returns. Therefore an important procedurebefore volatility analysis is to convert original prices toreturns. The definition for the rate of financial return Rg isthe logarithm of price ratio; that is Rg¼ ln Pt�ln Pt�1,where Pt and Pt�1 are the prices at time t and t�1,respectively. In our work the same definition is utilised indefining the rate of random electricity price returns. In thesame manner, the rate of load log-scale difference Dg isdefined as Dg¼ ln Lt�ln Lt�1. Because these randomsignals are the residuals after filtering out their periodicsignals, these residuals can sometimes be negative. Thus anessential procedure is to shift all the random values to bepositive to meet the foregoing mathematical function.

The rate of return for random electricity price in April2001 for the ISO-NE is illustrated in Fig. 10. Attention mustbe paid to the fact that the return series in Fig. 10 exhibits acharacteristic known as volatility clustering. That is, largechanges and small changes tend to have their own clusters.In addition, a sample for the rate of load log-scale differenceis shown in Fig. 11. In contrast to random price, thefluctuation of random load is low.

To evaluate the volatility for the random prices andrandom loads, we propose the GARCH (1,1) model tomeasure their volatility because these models are capable ofcapturing the volatility clustering observed in electricityprice returns. The generalised autoregressive conditionalheteroskedasticity (GARCH) model is often used forevaluating financial market volatility [17]. The term

‘heteroscedasticity’ represents a ‘changing variance’, andthe term ‘conditional’ indicates that the variance of time-series depends on past information. GARCH (p,q) modelsthe residual of a time-series regression. Let et denote a real-valued discrete-time stochastic process ct�1 be the informa-tion set of all information through time t-1, and ht be theconditional variance. The GARCH (p,q) process [17] can beexpressed as

et ct�1j � N 0; htð Þ ð8Þ

ht ¼ a0 þXq

j¼1aje2t�j þ

Xp

i¼1biht�i ð9Þ

where a040, ajZ0, and biZ0. The simplest but often veryuseful process is the GARCH (1,1) process when both p andq are set at one. A number of financial papers [18] havereported that a simple GARCH (1,1) model can capture thevolatility in most of financial returns. The typical GARCH(1,1) model consists of a conditional mean model and aconditional variance model based on (10) and (11),respectively. That is, the GARCH model is used tocharacterise the conditional distribution of random compo-nent et by imposing serial dependence on the pastconditional variance of the returns

yt ¼ gþ et et ct�1j � N 0; htð Þ ð10Þwhere

ht ¼ s2t ¼ a0 þ a e2t�1 þ b s2t�1 ð11ÞIn (11) the volatility s2t depends on a constant a0, the last

period’s variance s2t�1 and the last period’s innovation e2t�1(last period’s residual from the regression model); b is a keypersistence parameter: high b implies a high carry-overeffect of past to future volatility, while low b implies aheavily damped dependence on past volatility. Equation(11) is a recursive function and can be translated into (12). Itis clear from (12) that the volatility for time-series returnscan be determined by the sum of all past variances withcorresponding decaying factor bj. In other words, if the timeof variance in the past is close to the present its importancewould be increased.

s2t ¼a0 þ ae2t�1 þ b a0 þ ae2t�2 þ bs2t�2�

¼a0 þ ba0 þ b2a0 þ :::þ a e2t�1 þ be2t�2 þ b2e2t�3 þ :::�

¼ a01� b

þ aX1j¼1

bje2t�j

ð12ÞOn the basis of the GARCH (1,1) model, the time-varyingvolatility s2t on both prices and loads could be calculatedfrom the historical random component yt. The results forthe parameter estimations on GARCH (1,1) model aregiven in Table 7. These parameters and volatility series areestimated by using the maximum-likelihood algorithminside the GARCH toolbox. There is no direct measureof real volatility because volatility is inherently unobser-vable. In recent years some authors have proposed the useof realised volatility [18] as a proxy for real volatility, whichmeasures volatility without using any model. This algorithmuses high-frequency returns, such as hourly returns, to builddirectly an estimate of volatility at a lower frequency level,such as daily returns. However, it is not appropriate for ourapplications because of insufficient intra-hour data.

An indication of the success of GARCH models is thatthere is no autocorrelation in the squared standardisedreturns y2

t =s2t , which means the chosen model is capable of

2.0

1.5

1.0

0.5

−0.5

−1.0

−1.5

0

0 100 200 300 400 500 600

time, h

the

rate

of f

inan

cial

ret

urn

Fig. 10 Rate of electricity price returns in April 2001 for ISO-NE

0.20

0.10

−0.10

−0.200 100 200 300 400 500 600

time, h

the

rate

of l

oad

diffe

renc

e

0

Fig. 11 Rate of load log-scale difference in April 2001 forISO-NE

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 449

capturing volatility clustering for the original returns [19]because volatility clustering implies a strong autocorrelationin squared returns. Figure 12 shows the autocorrelationfunction of the squared standardised returns based on theISO-NE filtered price returns for the whole sample data in2001. It indicates this signal is uncorrelated and thereforeproves that the GARCH (1,1) model is well-specified. TheGARCH (1,1) model was also proven to be suitable forother filtered signals using the same diagnosis.

In Figs. 13 and 14 the thin lines show the rate of pricereturns and demand log-scale difference, respectively. Thecorresponding volatility measured by the proposedGARCH (1,1) model is shown by the thick line in bothfigures.

To evaluate the volatility relationship between randomloads and random prices we make use of the probabilitydistribution of correlation coefficients. These correlation

coefficients between the volatility of prices and loads areobtained from the volatility measure results based on theGARCH (1,1) model. In addition the whole sample data in2001 are utilised for this analysis. First, the correlationbetween the volatilities of random prices and random loadswas analysed for daily peak and off-peak periods. Theresults are presented in Table 8. It can be seen that a highercorrelation exists in the NEM-NSW market. Moreover,there is no conspicuous correlation divergence between peakand off-peak periods, which is different from the result forthe correlation between random loads and random prices asdescribed in Table 3.

Secondly, the correlation between the volatilities ofrandom loads and random prices based on different loadlevels who also examined. The time-varying volatilityfor random loads was divided into three levels and basedon the cum

ffiffiffiffif3p

rule. On the basis of the three differentlevels, we analysed the relationship between randomprice volatility and random load volatility with theprobability distribution of correlation coefficients. Theprobability that correlation coefficients are larger than0.8 and 0.6 is examined, and the statistical results shown inTable 9. It is observed form Table 9 that high-load volatilitycan lead to high price volatility. On the other hand, thecorrelation between load volatility and price volatility isvery low in the low strata. This clearly supports thenonexisting correlation between load and price volatilities inthe low-load volatility level.

Based on the results of Tables 1, 3, 8, and 9, theprice–load relationship in the NEM is higher thanthat in the ISO-NE in both periodic and nonperiodic

Table 7: Parameter estimations on GARCH (1,1) model forvolatilities of random prices and random loads

Powermarket

Series type Meanequation

Variance equation

g a0 b a

ISO-NE load difference 0.0025 0.0013 0.3565 0.4173

price returns �0.00016 0.0056 0.5433 0.3309

NEM-NSW load difference 0.00091 0.0001 0.5499 0.4444

price returns 0.0017 0.0016 0.3511 0.5477

0.8

0.6

0.4

0.2

−0.2

0

0 5 10 15 20 25 30 35 40

lag

auto

corr

elat

ion

coef

ficie

nt

Fig. 12 Autocorrelation function of squared standardised returnsbased on ISO-NE filtered prices

0.6

0.4

0.2

−0.2

−0.4

−0.6

−0.8

0

0 100 200 300 400 500 600

time, h

the

rate

of p

rice

retu

rns

and

corr

espo

ndin

g vo

latil

ity

Fig. 13 Rate of random price returns and corresponding volatilityfor ISO-NE, October 2001

0.20

0.15

0.10

0.05

−0.05

−0.10

−0.15

−0.200 100 200 300 400 500 600

time, h

the

rate

of l

oad

diffe

renc

e an

d co

rres

pond

ing

vola

tility

0

Fig. 14 Rate of random load log-scale difference and correspond-ing volatility for ISO-NE, October 2001

Table 8: Correlation analysis between random price andrandom load volatilities for different periods

Powermarket

Time section Probability, %(correlationcoefficient40.6)

Probability, %(correlationcoefficient40.8)

ISO-NE peak hourperiods

12.80 2.4

off-peakhour periods

20.83 10.0

NEM-NSW peak hourperiods

48.00 8.00

off-peakhour periods

43.33 18.33

450 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

components. It appears that the regional prices calculatedduring each half-hour period provide more stable pricesignals, which have a high price–load relation feature. Thiscould be one of the reasons why the recent development ofpower market designs has shown a trend moving fromuniform pricing (such as the UK electricity pool and thepre-2003 ISO-NE market) and regional pricing (such asthe NEM and ERCOT market) to fully locational pricing(such as the PJM, NY-ISO, and the post-2003 ISO-NEmarket).

7 Conclusions

Analysis of the relationship between electricity prices andloads is currently receiving a great deal of attention. In thecase of periodic components this paper provided empiricalsupports for the view that identical bin numbers appear inthe spectra of prices and loads. The deterministic correlationbetween prices and loads could be decided by these binnumbers. In the random components it is obvious thathigher price–load correlation exists in peak hours based onour statistical analysis. In addition, for the volatility relationanalysis between prices and loads the results show that thereis no conspicuous correlation divergence between peak andoff-peak periods. Nevertheless, electricity price volatilityinherits high-load volatility, but not in the case of low-loadvolatility.

The results of the analysis of the price–load relationshipwould change based on different market conditions. Forexample, long-term contracts, high available capacity,transfer capability and spinning reserves, and high stabilityof fuel prices would contribute to a stable price trend, whichwould form a strong relationship between prices and loads.In this situation the periodic components on prices wouldincrease. However, unstable physical factors would lead to alow relationship between prices and loads, which meansthat the ratio of random prices would increase. In otherwords, the ratio of periodic components on prices could beregarded as an index to evaluate the market price stability.

This paper has provided a quantitative analysisbetween price and demand. Its aim has been to highlighta simple and useful statistical approach to discover theprice–load relationship. Market traders could incorporatethe statistical results and load forecasting into the pricingmodule to predict electricity price. For example, duringpeak hours the results of load forecasting could be takeninto account for price forecasting by using time-seriestransfer functions or ANN because higher price–loadcorrelation exists in this period. A similar procedurecould also be applied to price volatility forecastingduring high-load volatility. In addition, the results couldsupport market traders to make useful decisions withregards to their bids and offers in financial hedging

markets. For example, based on the statistical results theVAR is proportional to power loads, which encouragestraders to purchase long-term forward contracts duringhigh-load periods to hedge against the risk of high prices.Moreover, power suppliers for residential customerscould refer to these empirical statistics to implement real-time pricing, which might bring the price-elasticity ofdemand and reduce the need for generation capacity in thepower system.

The proposed approach is based on direct statisticalanalysis. The contribution of this research is to provide asimple and useful approach to evaluate the price–loadrelationship. In addition, it presents a line of inquiry thatmay lead to further analyses as follows.

� The relationship analysis between prices and loads can beextended to the combined time–frequency analysis, such asGabor or wavelet. This analysis provides a time-varyingspectrum, which allows one to determine which frequenciesexist at a particular time.

� It is useful to incorporate the GARCH stochasticvolatility into the options model of electricity prices, whichcould replace the traditional Black–Scholes valuationmodel.

� By reviewing the price stability index it may give systemoperators some suggestions to improve their marketdesigns.

8 References

1 Cheung, K.W., Sun, D., Milligan, J., and Potishnak, M.: ‘Energyand ancillary service dispatch for the interim ISO New Englandelectricity market’, IEEE Trans., Power Syst., 2000, 15, (3), pp.968–974

2 Kian, A., and Keyhani, A.: ‘Stochastic price modeling of electricity inderegulated energy markets’. Proc. 34th Hawaii Int. Conf. on SystemSciences, Hawaii, USA, 3–6 Jan., 2001, pp. 832–838

3 Yang, J., and Anderson, M.D.: ‘Dynamic pricing [of electricity]’,IEEE Potentials, 2000, 18, (5), pp. 6–9

4 Jinxiang, Z., Jordan, G., and Ihara, S.: ‘The market for spinningreserve and its impacts on energy prices’. Proc. IEEE PowerEngineering Society Winter Meeting, Singapore, 23–27 Jan., 2000,Vol. 2, pp. 1202–1207

5 Vucetic, S., Tomsovic, K., and Obradovic, Z.: ‘Discovering price–loadrelationships in California’s electricity market’, IEEE Trans., PowerSyst., 2001, 16, (2), pp. 280–286

6 Niimura, T., and Nakashima, T.: ‘Deregulated electricity market datarepresentation by fuzzy regression models’, IEEE Trans., Syst., Man,Cybern., 2001, 31, (3), pp. 320–326

7 Mount, T.: ‘Market power and price volatility in restructured marketsfor electricity’. Proc. 32nd Hawaii Int. Conf. on Systems Sciences,Hawaii, 5–8 Jan., 1999, pp. 13–15

8 Managing energy price risk, (Risk Books, London , 1999, 2nd Edn.)9 Mulgrew, B., Grant, P., and Thompson, J.: ‘Digital signal processing

Concepts and applications’ (Palgrave, Basingstoke, 2002)10 Oran Brigham, E.: ‘Fast Fourier transform and its applications’,

(Prentice Hall, 1988, 1st Edn.)11 Cochran, W.G.: ‘Sampling techniques’ (Wiley, Englewood Cliffs,

NJ, 1997)12 Singh, R.: ‘Approximately optimum stratification on the auxiliary

variable’, J. Am. Stat. Assoc., 1971, 66, pp. 829–833

Table 9: Correlation analyses between random price volatility and random demand volatility according to different demandstrata

Power market Random demandstratum

Threshold region Probability, % (correlationcoefficient40.6)

Probability, % (correlationcoefficient40.8)

ISO-NE high large than 0.2104 66.67 0

medium between 0.2104 and 0.0721 17.50 12.50

low less than 0.0721 4.88 0

NEM-NSW high large than 0.0576 80 80

medium between 0.0576 and 0.0283 20.00 8.00

low less than 0.0283 0 0

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 451

13 Alexander, C.: ‘Risk management and analysis, Vol 1: Measuring andmodelling financial risk’, (Wiley, New York, 1999)

14 Cohen, L.: ‘Time–frequency analysis’, (Prentice Hall, EnglewoodCliffs, NJ, 1995, 1st Edn.)

15 Jarrow, R.: ‘Volatility: new estimation techniques for pricingderivatives’ (Risk Books, London, 1998)

16 Fouque, J.-P., Papanicolaou, G., and Sircar, K.R.: ‘Derivatives infinancial markets with stochastic volatility’ (Cambridge UniversityPress, New York, 2000)

17 Bollerslev, T.: ‘Generalized autoregressive conditional heteroskedasti-city’, J. Econometr., 1986, 31, pp. 307–327

18 Ser-Huang Poon, ., Clive, W.J., and Granger, .: ‘Forecastingvolatility in financial market: a review’, J. Econ. Lit., 2003, XLI,pp. 478–539

19 Alexander, C.: ‘Market models: a guide to financial data analysis’(Wiley, New York, 2001)

9 Appendix

Proof of DFT SymmetryTo prove the symmetry properties of the DFT of a sampleddata sequence x(n) one could substitute k¼N�r into

the DFT (1), where roN/2. Then, (1) could be translatedinto (13)

X N � rð Þ ¼XN

n¼1x nð Þ exp �j n� 1ð Þ N � r � 1ð Þ2p

N

� �

¼XN

n¼1x nð Þ exp j n� 1ð Þ r þ 1ð Þ2p

N

� �

exp �j n� 1ð Þ2pð Þ

¼XN

n¼1x nð Þ exp j n� 1ð Þ r þ 1ð Þ2p

N

� �

ð13Þ

Therefore X(N�r)¼X*(r+2)

452 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004