or

simcludeeklf ad. Oyoa Mingnd o

Introduction

ents aies, eno optimrationonths

rage wt categal variehavioine [2]

analytics service provider Point Carbon [6] utilizes a stochasticdual dynamic programming (SDDP) model which they claim yieldsimproved accuracy compared to the EMPS model. However theyprovide no numerical results to back up their statements. In sum,these models may appear as black box approaches to outsiderswhich do not have access to the inputs. Therefore they are unableto adjust these if necessary.

In this paper we develop three relatively simple spot price mod-

Auto-regressive (AR) models. Auto-regressive moving average (ARMA). Auto-regressive integrated moving average (ARIMA) and sARIMA models [8,9].

Auto regressions with heteroskedastic [10]. Heavy-tailed innovations [11]. AR models with exogenous (fundamental) variables. Dynamic regression (or ARX) and transfer function (or ARMAX)models [12].

Vector auto regressions with exogenous effects [13]. Tel.: +45 201144160.

E-mail address: tarjeikr@yahoo.com

Electrical Power and Energy Systems 61 (2014) 2026

Contents lists availab

n

.e lthe model are proprietary and differ for each market player makingit difcult to facilitate a comparison. Likewise the power market

Time series models include:http://dx.doi.org/10.1016/j.ijepes.2014.03.0070142-0615/ 2014 Elsevier Ltd. All rights reserved.ations.

easonalchine [3], auto-regressive fractionally integrated moving average[4] and Grey models [5].

The most commonly applied model for medium term spot priceforecasting in the Nord Pool market is the EMPS model (or Sam-kjringsmodellen). It is a fundamental stochastic dynamic pro-gramming model for hydro-thermal scheduling. The results from

Weron [7] analyzes forecasting of day-ahead electricity pricesbut nds that only some approaches are suitable. Time series mod-els are among the models with strongest forecasting ability. Weron[7] also identies that single hour model specications achievebetter forecasting accuracy than multi-hour model specicShort-term price forecasting presprotability of generation compantrading companies. It enables them tegies. The planning horizon for genenies typically spans weeks and/or mfor models that can forecast the aveseries models represent an importanmodels typically utilize some externand/or demand to explain pricing bclude multiple support vector machcrucial activity for thed-user companies andize their trading strat-and consumer compa-. Thus there is a needeekly spot prices. Timeory of models [1]. Theseables related to supplyr. Other approaches in-, extreme learning ma-

els that utilize previous weeks spot prices, futures prices includingdata for inow and reservoir levels. The models provide relativehigh precision spot price forecasts for the Nord Pool market.

The structure of the paper is as follows. Section Literaturereview provides a literature review. Section The Nordic powermarket describes the Nordic power market and its fundamentalparameters. Section Spot forecasting models develops three spotprice forecasting models. Section Model performance measuresthe model performances. Section Conclusion concludes.

Literature reviewA time series spot price forecast model f

Tarjei Kristiansen 9000 lborg, Denmark

a r t i c l e i n f o

Article history:Received 11 January 2013Received in revised form 16 February 2014Accepted 7 March 2014Available online 2 April 2014

Keywords:Nord poolSpot price forecastingRegression modelElectricity market

a b s t r a c t

We present three relativelyspot and futures prices inaccurate forecast of the wpercentage error (MAPE) oMW h in the sample perioof the actual spot price. A mweeks spot price achievesMW h. A futures model usachieves a MAPE of 5.3% a

Electrical Power a

journal homepage: wwwthe Nord Pool market

ple spot price forecast models for the Nord Pool market based on historicing data for inow and reservoir levels. The models achieve a relativelyy spot prices. The composite regression model achieves a mean absoluteround 7.5% and under-forecasts the actual spot price by some 1.4 NOK/ut of sample testing achieves a MAPE of around 7.4% including a matchpic model using the previous weeks spot price as a predictor for the nextAPE of 7.5% and under-forecasts the actual spot price by some 0.9 EUR/the futures price for next week as a predictor for next weeks spot pricever-forecast the actual spot price by some 4.3 EUR/MW h.

2014 Elsevier Ltd. All rights reserved.

le at ScienceDirect

d Energy Systems

sevier .com/locate / i jepes

Threshold AR and ARX models [14]. Regime-switching regressions with fundamental variables [15]. Mean-reverting jump diffusions [16].

Kristiansen [17] developed an auto-regressive ARX model forNord Pool day ahead prices. The model is based on Weron andMisiorek [11] but reduced in terms of estimation parameters(from 24 sets to 1) and modied to include Nordic demand andDanish wind power as exogenous variables. Prices are modeledacross all hours in the analysis period rather than across each sin-gle 24 h. By applying three model variants on Nord Pool data, aweekly mean absolute percentage error (WMAE) of around 67% and an hourly mean absolute percentage error (MAPE) rangingfrom 8% to 11% were achieved. Out of sample results yielded aWMAE and an hourly MAPE of around 5%. Kristiansen [18] alsodeveloped a fundamental stack model for the German electricitymarket.

Weron and Misiorek [11] achieved a weekly-weighted meanabsolute error of 4.66% for four ve-week periods (i.e. 20 weeks)from 1998 to 1999 and 3.37% for four ve-week periods (i.e.20 weeks) from 2003 to 2004. Kristiansens [17] weekly-weightedmean absolute errors are somewhat larger but he has formulateda different model and included a larger and more recent data per-iod so results are not directly comparable.

Torro [1] developed an ARIMAX model to forecast weekly fu-tures prices in the Nord Pool market for the period 19972003.The time-series model contains external variables such as temper-ature, precipitation, reservoir levels and the basis (futures priceminus the spot price) which, overall, reects the typical seasonalpatterns in the weekly spot price. Torro [1] achieved an average er-ror of 0.78 NOK/MW h for one week ahead futures but providedno model for spot price forecasting. Torro [1] also proposed twoalternative and simpler forecasting methods:

A myopic method that utilizes the present spot price as a fore-cast for next week. Torro [1] explains that a myopic method canbe considered as the minimum accuracy deemed from any fore-casting method.

A futures method that utilizes the present futures price as aproxy for next weeks spot price.

We utilize these models as benchmarks in this paper.

The Nordic power market

The Nordic power exchange, Nord Pool, was established in 1993and consisted of Norway, subsequently joined by Sweden in 1996.

pot p

T. Kristiansen / Electrical Power and Energy Systems 61 (2014) 2026 21Fig. 1. Average, minimum and maximum weekly sFig. 2. Average, minimum and maximum reservoir levelsrices in the Nord Pool market from 1999 to 2010.in percentage in Norway for the period 19992010.

els i

22 T. Kristiansen / Electrical Power and Energy Systems 61 (2014) 2026Finland in September 1998, Western Denmark in January 1999,and eastern Denmark in October 2000. Nord Pool operates the spotmarket [19] while NASDAQ OMX operates the nancial market.The daily spot market is cleared the day before operation basedon the market players offers to sell or bids to buy. The system priceis the price for Nordic market ignoring transmission constraints. Itis used as a settlement price for the forward and futures markets.Transmission congestion causes area prices to differ within and be-tween the Nordic countries.

Approximately 50% of the Nordic electricity is generated fromhydropower plants. Norway has close to 100% hydro while Swedenand Finland have around 50% and 20% share, respectively. Themajority of the hydropower is hydro storage with potential formulti-year storage in Norway. Hydro storage and demand inu-ence both the short- and long-term prices. Hydro storage levelspeak in SeptemberNovember and reach their lowest levels inAprilMay. Conversely, demand levels peak in the winter (Decem-berMarch) due to demand for electric heating. Electricity pricesthus have some seasonality and reach their lowest levels in theperiod MayAugust. Healthy hydro supplies facilitate fast rampingand exibility features. During demandsupply shocks prices areless volatile compared to a pure thermal power system. Howeverduring drought and ooding situations volatility may increase.

Fig. 3. Average, minimum and maximum weekly inow levThe historic system spot price and hydro storage levels in Nor-way exhibited a seasonal pattern during the period 19962001 asshown in Fig. 1. However, the seasonal pattern weakened after2001.

There is a relationship between hydro storage and the spot pricelevels. Norway has the dominant share of storage capacity(84.3 TWh) in the Nordic power system. The major price increasesin the fall/winter 2002/2003 and summer 2006 reected low reser-voir levels (see Fig. 2). A similar low reservoir level in 1996/97 alsoled to an increase in price. The period 20022006 was on average

Table 1Correlation between the natural logarithm of the de-meaned weekly spot prices in weekSweden in GWh and the natural logarithm of change in reservoir level in week t 1 minu

ln average spotweek t

ln average spot weekt 1

ln average spot week t 1ln average spot week t 1 0.97 1ln inow (NO + SWE) (GWh) week

t 10.32 0.32

ln change in reservoir levels % weekt 1

0.21 0.22drier than the period 19962001 (i.e. lower reservoir levels). Theperiod 20072010 included high reservoir levels for 2007 and2009 while 2010 had low reservoir levels.

The relationship between reservoir levels and electricity futuresprices at Nord Pool has been studied by Gjolberg and Johnsen [20]and Botterud et al. [21]. The authors consistently found that thereservoir level and its seasonality dependence largely explain theprice formation for spot and future prices. Storage theory statesthat seasonality generates seasonal patterns in spot and futureselectricity prices [1]. Reservoir levels impact the sensitivity of de-mand and supply shocks. A shock occurring at a high reservoir le-vel is uncritical. Conversely a shock at a low reservoir level isharder to match and may cause rising prices. Full reservoirs maycause spill and reduced prots. Thus a negative convenience yield(i.e. high reservoir levels) may cause producers to sell at lowerprices rather than risking spill [1]. The result would be lower spotprices than futures prices. Conversely a positive convenience yield(i.e. low reservoir level), may cause higher spot prices than futuresprices. Likewise, precipitation expectations inuence the powerprice formation.

The inow levels to reservoirs from snowmelting peak betweenweeks 18 and 27 as shown in Fig. 3. Conversely during the winter,inows are at their lowest levels for the weeks 117.n GWh for Norway and Sweden for the period 19992010.Spot forecasting models

In the following we propose three models to forecast averageweekly spot price in the Nord Pool market. The rst model assumesthat the average weekly spot price in the current week depends onthe average weekly spot price in the previous week (auto-correla-tion), the inow level in Norway and Sweden in the previous weekin GWh and the difference in reservoir level in Norway in the two

t and week t 1, the natural logarithm of inow levels in week t 1 in Norway ands week t 2 in Norway in percent of maximum reservoir level.

ln inow (NO + SWE) (GWh) weekt-1

ln change in reservoir levels % weekt 1

1

0.91 1

preceding weeks as percentage of maximum reservoir level. Wehave calculated the natural logarithm of all variables. Furthermore,we have de-meaned the price time series by removing the meanof log return from the log returns such that the average is zero. Wehave performed this operation to facilitate that the process underconsideration is a stationary stochastic process. We will test forthis issue later in this paper but the mean of the Nordic electricityprice is not constant across time but exhibits seasonality.

These variables exhibit the correlation shown in Table 1. Thereis a high correlation between the spot price in the current weekand the previous week. There is a weaker negative correlation be-tween spot prices and inow levels. This reects that a higher in-ow level improves the supply situation and causes prices todecline. Conversely a lower inow level deteriorates the supply sit-uation and causes price to incline. Likewise there is a weak nega-tive correlation between the spot price and the change in the

where A is actual value and F is forecast value.The performance and accuracy of the price forecasting models

are summarized in Table 5. Likewise we have charted the forecast-ing models against the actual spot price in Fig. 4.

Table 4Augmented Dickey Fuller (ADF) test for the period 19992010.

Time series Ln de-meaned spot price 19992010

Dickey Fuller test statistic 5.4914

1 Stationarity, is dened as a quality of a process in which the statistical parameters(mean and standard deviation) of the process do not change with time. The mostimportant property of a stationary process is that the auto-correlation function (ACF)depends on lag alone and does not change with the time at which the function was

T. Kristiansen / Electrical Power and Energy Systems 61 (2014) 2026 23Norwegian reservoir levels in the two preceding weeks. A deterio-rating reservoir level causes the price to increase. Conversely animproving reservoir level causes the price to decrease.

To test for multicollinearity we used the Variance Ination Fac-tor (VIF) that measures multicollinearity in the model. VIF mea-sures the magnitude of the change of the estimated coefcientsover the no correlation case among the other variables. If no vari-ables are correlated the VIFs will be 1. If the VIF is around 4 orgreater for one variable, then one independent variable is highlycorrelated with the other variables.

We regressed the variables the natural logarithm of the spotprice in week t and week t 1 (ln S t and ln S t 1), the natural log-arithm of the inow in week t 1 (ln inf t 1) and the natural log-arithm of the change in reservoir level from week t 2 to weekt 1 (ln delta res t 1) against each other and calculated theVIF = 1/(1R2). The VIF associated with the regression of ln deltares t 1 and ln S t 1 against ln inf t 1 was 3.55, the VIF associ-ated with the regression of ln inf t 1 and ln S t 1 against ln deltares t 1 was 3.36 while the VIF associated with the regression of lndelta res t 1 and ln inf t 1 against ln S t 1 was 1.02. Thus weconclude there is no substantial muliticolinarity .

The model takes the following functional form:

st c a st1 b it1 d rt1 etwhere c is the intercept of the regression equation, st = ln(St) is thenatural logarithm of the weekly spot price in the current week,st1 = ln(St1) is the natural logarithm of the weekly spot price inthe previous week, it1 = ln(It1) is the natural logarithm of inowlevel in the previous week in GWh in Norway and Sweden,rt1 ln Rt1Rt2

is the ratio between the reservoir levels in Norway

Table 2Regression statistics for the auto-regressive model.

Regression statistics

Multiple R 0.9746R Square 0.9498Adjusted R square 0.9495Standard error 0.1056Observations 581

Table 3Regression coefcients and associated statistics for the auto-regressive model for the19992010 data period.

Coefcients Standard error t Start p-Value

Intercept 0.3285 0.1122 2.9279 0.0035ln S t 1 0.9684 0.0101 96.3215 0.0000

ln inf t 1 0.0412 0.0142 2.9038 0.0038ln delta res t 1 0.5858 0.2027 2.8899 0.0040in the previous week and the preceding week in percent of maxi-mum reservoir level, a, b, d are regression coefcients and et is as-sumed to be independent and identically distributed with zeromean and nite variance.

To estimate the parameters of the suggested model we haveobtained historical data for weekly spot prices, inow levels inNorway and Sweden and reservoir levels in Norway for the periodfromweek 44 in 1999 to week 52 in 2010. The results of the regres-sion are shown Tables 2 and 3. The R square equals 0.97 and isrelatively high. In regression analysis, the t-stat, coupled with itsp-value, indicates the statistical signicance of the relationship be-tween the independent and dependent variable. All p-values arebelow 0.05 and indicates that the selected variables provide a suit-able description of the spot price in the current week.

The characteristics for a stationary time series are a constantmean and variance. Therefore a time series with trend is non-stationary. Table 4 shows the augmented dickey fuller (ADF) testfor stationary1 conditions for the time series from 1999 to 2010.The critical values for the ADF test based on F-Statistic are 3.98for p = 0.01 and 3.68 for p = 0.025 when the number of observa-tions is equal to 500. The ADF test gives p = 0.01 and a Dickey Fullertest statistic of 5.4914.

We calculated the Hurst exponent to 0,9775 which indicatesthat the time series have persistence or long term positive auto-correlation.

In addition we utilize the models proposed by Torro [1] asbenchmarks:

A myopic method that utilizes the present spot price as a fore-cast for next week. Torro [1] explains that a myopic method canbe considered as the minimum accuracy required from anyforecasting method

A futures method that utilizes the closing price of futures for thelast workday in the preceding week as a proxy for next weeksspot price.

Model performance

To measure the performance of the models we have used themean absolute percentage error (MAPE) dened as the averageabsolute difference between the actual value and the forecast valuedivided by the actual value:

MAPE 1n

Xnt1

At FtAt

p-Value 0.01Lag order 15calculated. A weakly stationary process has a constant mean and ACF (and thereforevariance). A truly stationary (or strongly stationary) process has all higher-ordermoments constant including the variance and mean.

The regression model achieves a mean absolute percentage er-ror of 7.5% and under-forecasts the spot price by 1.4 NOK/MW h.This result is comparable with what Kristiansen [17] achieved witha weekly mean absolute percentage error (WMAE) of around 67%.However the price deviation was somewhat higher and in therange 1.72.6 NOK/MW h.

The myopic model achieves a MAPE 7.5% and over-forecasts thespot price by some 0.9 NOK/MW h. The futures model achieves aMAPE 5.3% and over-forecasts the spot price by some 4.3 NOK/MW h.

The futures model achieves the best performance in terms ofMAPE equal to 5.3% while the regression and myopic models havethe highest MAPEs of 7.5%. The deviation is lowest for the myopicand regression models. The absolute deviation is lowest (14.2 NOK/MW h) for the futures model and highest for the myopic model(19.9 NOK/MW h).

Table 5Weekly MAPE and absolute price deviation (NOK/MW h) for the three proposedmodels.

Model MAPE Deviation forecastand spot price(NOK/MW h)

Average absolute deviationforecast and spot price(NOK/MW h)

Regressionmodel

7.5% 1.4 19.6

Myopicmodel

7.5% 0.9 19.9

Futuresmodel

5.3% 4.3 14.2

astin

Table 6Regression coefcients and associated statistics for the 19992006 data period whichis used for the out sample model testing.

Coefcients Standard error t Stat p-Value

Intercept 0.3888 0.1380 2.8179 0.0051ln S t 1 0.9589 0.0128 74.8375 0.0000ln inf t 1 0.0497 0.0175 2.8340 0.0048ln delta res t 1 0.6289 0.2483 2.5332 0.0117

Table 7Weekly MAPE and absolute price deviation (NOK/MW h) for the out of sample period.

MAPE Deviation forecast and spot price (NOK/MW h)

Model 7.4% 0.0

24 T. Kristiansen / Electrical Power and Energy Systems 61 (2014) 2026Fig. 4. Comparison of price forecFig. 5. Autocorrelation function for the residual (condence interval in red). (For interpreweb version of this paper.)g models and actual spot price.tation of the references to colours in this gure legend, the reader is referred to the

nd ET. Kristiansen / Electrical Power aThus we can conclude that a pure myopic model which utilizesthe previous weeks spot price as a forecast for the current weekhas a similar performance as the regression model. However thefutures model has the best overall performance in terms of MAPEand absolute price deviation but it over-forecasts the spot price.It is worth noting that the futures model has some large deviationsfor the end of 2002 which was a period with drought. Also for the

Fig. 6. Partial autocorrelation function for the residual (condence interval in red). (For ito the web version of this paper.)

Fig. 7. Autocorrelation function for the squared residual (condence interval in red). (Forto the web version of this paper.)

Fig. 8. Partial autocorrelation function for the squared residual (condence interval in rereferred to the web version of this paper.)nergy Systems 61 (2014) 2026 25spot and the regression models there were some larger deviationsfor the end of 2002.

To further test the regression model we performed out of sam-ple test by estimating the regression coefcients in Table 6 fromthe period week 44 in 1999 to week 52 in 2006. The R squarewas 0.97. We subsequently applied the regression parameters todata for the period from 2007 to 2010.

nterpretation of the references to colours in this gure legend, the reader is referred

interpretation of the references to colours in this gure legend, the reader is referred

d). (For interpretation of the references to colours in this gure legend, the reader is

The performance of the out of sample test is shown in Table 7.The performance of the MAPE is similar to the previous test. How-ever the deviation between the forecast and the spot price is some-what lower.

For a given stochastic process one is often interested in the con-nection between two random variables of a process at differentpoints in time. One way to measure a linear relationship is withthe auto-correlation function (ACF), which measures the correla-tion between these two variables. We have applied the ACF withvarious time lags on the residuals and the squared residuals inFigs. 58. We note the alternating and tapering patterns for theresidual and the squared residuals. For the residuals both chartsexhibit a similar pattern. The ACF and PACF functions show a def-inite pattern with decreasing lags and thus a trend in the data. Themagnitude of the ACF and PACFs are relatively small but the lagsare larger at orders 1, 2, 4, 5, 10 and 16 for the ACF and at orders1, 2, 3, 5 and 16 for the PACF. Also the squared residuals ACF andPACF functions exhibit a similar pattern with decreasing lags.The ACF tails off around 6 lags while the PACF tails off after 2 lagsbut also have one spike at lag 5.

model is a nave model by assuming an extrapolation of the currenttrend. The regression model has a similar feature but also utilizesfundamental information such as reservoir and inow levels.

References

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day electricity prices. IEEE Trans Power Syst 2003;18(3):101420.

26 T. Kristiansen / Electrical Power and Energy Systems 61 (2014) 2026Conclusion

This paper has presented three relatively simple spot priceforecasting models for the Nord Pool market. The regression modelrequires historic spot prices, inow and reservoir levels for theestimation of the regression coefcients. The model achieves aR-square of around 0.97. The mean absolute percentage error is 7.5%and it under-forecasts the actual spot price with 1.4 NOK/MW h.Out of sample achieves a mean absolute percentage error of 7.4%and matches the actual spot price.

The myopic model based on historic spot prices by extrapolat-ing the current weeks spot price as a predictor for next weeksprice. Its MAPE is 7.5% and the price deviation is 0.9 NOK/MW h. The futures price model which takes the last futures closingprice for the week as a predictor for next weeks spot price, has aMAPE of 5.3% and deviation of 4.3 NOK/MW h.

We conclude that the futures model outperforms the othermodels with a lower MAPE and average absolute price deviation.The regression and myopic models have a similar performance.The last futures closing price for the week ahead contract shoulddiscount all available information to the market. Thus it could beconsidered as a price forecast for the following week. The myopic[9] Zhou M, Yan Z, Ni Y, Li G, Nie. Electricity price forecasting with condence-interval estimation through an extended ARIMA approach. IEE Proc GenerTrans Distrib 2006;153(2):2338.

[10] Garcia RC, Contreras J, van Akkeren M, Garcia JBC. A GARCH forecasting modelto predict day-ahead electricity prices. IEEE Trans Power Syst2005;20(2):86774.

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[13] Panagiotelis A, Smith M. Bayesian forecasting of intraday electricity pricesusing multivariate skew-elliptical distributions. Int J Forecast 2008;24(4):71027.

[14] Misiorek A, Truck S, Weron R. Point and interval forecasting of spot electricityprices: linear vs. non-linear time series models. Nonlinear Dyn Econom2006;10(3):2.

[15] Karakatsani N, Bunn D. Forecasting electricity prices: the impact offundamentals and time-varying coefcients. Int J Forecast 2008;24(4):76485.

[16] Knittel CR, Roberts MR. An empirical examination of restructured electricityprices. Energy Econ 2005;27(5):791817.

[17] Kristiansen T. Forecasting Nord Pool day-ahead prices with an auto-regressivemodel. Energy Policy 2012;49:32832.

[18] Kristiansen T. Power trading analytics and forecasting Germany. Electr J2011;24(8):4155.

[19] Nord Pool. 2012. .[20] Gjolberg O, Johnsen T. Electricity futures: inventories and price relationships

at Nord Pool. Discussion Paper D-16/2001, IS. 2001.[21] Botterud A, Kristiansen T, Ilic M. The relationship between spot and futures

prices in the Nord Pool electricity market. Energy Econ 2010;32(5):96778.

A time series spot price forecast model for the Nord Pool marketIntroductionLiterature reviewThe Nordic power marketSpot forecasting modelsModel performanceConclusionReferences