# DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON

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2 Next, the descrition of the general statistical methodology to build a final model is resented. The three models have been obtained through the following scheme: Ste. A class of models is formulated assuming certain hyotheses. Ste 1. A model is identified for the observed data. Ste 2. The model arameters are estimated. Ste 3. If the hyotheses of the model are validated go to Ste 4, otherwise go to Ste 1 to refine the model. Ste 4. The model can be used to forecast. In the following subsections, each ste of the above scheme is detailed. 2.1 Ste The building of each of the three models resented differs in this ste. This ste is exlained below deending on the selected model Dynamic regression The first roosed method to forecast rices is a dynamic regression model [, 11]. In this model, the rice at hour t is related to the values of ast rices at hours t 1, t 2,, etc. This is done to obtain a model that has uncorrelated errors. In Ste, the selected model used to exlain the rice at hour t is the following: where ( t c ω B t + εt = + (1 is the rice at time t and c is a constant. t Function = l K l ω ( B = 1ωl B is a olynomial function l of the backshift oerator B : Bt = t l, where the total number of terms of the function, K, is subject to change in stes 1-3. Function ω ( B deends on arameters ω l, whose values are estimated in Ste 1. Finally, ε t is the error term. In Ste this term is assumed to be a series drawn randomly from a normal 2 distribution with zero mean and constant variance σ, that is, a white noise rocess. The efficiency of this aroach deends on the election of the aroriate arameters in ω (B to achieve an uncorrelated set of errors. This selection is carried out through Stes 1 to 3 as it is exlained below Transfer function A second roosed method that includes a serially correlated error is called transfer function model [, 11]. Secifically, it is assumed that the rice and demand series are both stationary (i.e. with constant mean and variance. The general form roosed to model the (rice, demand transfer function is ( d t = c+ ω B dt + Nt (2 where t is the rice at time t, c is a constant, d t is d the demand at time t, = l K d l ω ( B = ωl B is a olynomial function of the backshift oerator, and N t is a disturbance term that follows an ARMA model of the form θ ( B Nt = εt (3 φ( B with θ Θ ( B l = 1 l θ =1 l B and φ = Φ ( B l 1 l φ =1 l B, both of which being olynomial functions of the backshift oerator. The total number of terms of the functions θ ( B and φ ( B, Θ and Φ, resectively, are subject to change in stes 1-3. Finally, ε t is the error term, that is assumed to be a white noise rocess. The model in (2 relates actual rices to demands d through function ω (B and actual rices to ast rices through function φ (B ARIMA The third roosed method is an ARIMA formulation. The roosed general ARIMA formulation in Ste is the following: ( B ( B φ t = θ ε t (4 where φ (B and θ (B are functions of the backshift oerator such as in (3, and ε t is the error term. But in this case, functions φ(b and θ (B have secial forms. They can contain factors of olynomial functions of the form φ Φ ( B l = 1 l φ =1 l B Θ l and/or θ ( B = 1 l θ =1 l B, and/or (1 B s, where several values of φ l and θ l can be set to. The total number of terms of the functions θ ( B and φ ( B, Θ and Φ, resectively, are subject to change in stes 1-3. For examle, function φ (B could have the following form: ( B = ( B B ( B B φ 1 φ1 φ2 1 φ24 φ ( 1 φ168b 1 ( B1 ( B ( It should be noted that this examle does not corresond to a standard ARIMA formulation, as resented in []. However, the model in (4 is sufficiently general to include the main characteristics of the rice data. For

3 examle, to include multile seasonality, factors of the form ( 1 φ24b, ( 1 φ168b, and/or ( 1 θ24b, ( 1 θ168b, and erhas (1 B, (1 B can be added to the model. It can be observed that the model in (4 relates actual rices to ast rices through function φ (B, and actual errors to ast errors through function θ (B. Finally, certain hyotheses on the three models must be assumed. These hyotheses are imosed on the error term, ε t. In Ste, this term is assumed to be a randomly drawn series from a normal distribution with zero 2 mean and constant variance σ. In Ste 3, a diagnostic checking is used to validate these model assumtions, as exlained in subsection Ste 1 Deending on the selected aroach, a trial model must be identified for the rice data. First, in order to make the underlying rocess stationary (more homogeneous mean and variance, a transformation of the original rice data may be necessary. In this ste, if a logarithmic transformation is alied to the rice data, a more stable variance is attained for the three models. For the dynamic regression and the transfer function aroaches, all the initial arameters are set to zero. Particularly, in the ARIMA formulation, the inclusion of factors of the form (1 s B may be necessary to make the rocess more stationary. And, to attain a more stable mean, factors of the form (1 B, (1 B, (1 B may be necessary, deending on the articular tye of electricity market, as exlained at the end of this section. The initial selected arameters for the ARIMA formulation are based on the observation of the autocorrelation and artial autocorrelation lots. In successive trials, the same observation of the residuals obtained in Ste 3 (observed values minus redicted values can refine the structure of the functions in the model. 2.3 Ste 2 After the functions of the models have been secified, the arameters of these functions must be estimated. Good estimators of the arameters can be found by assuming that the data are observations of a stationary time series (Ste 1, and by maximizing the likelihood with resect to the arameters []. The SCA System [12] is used to estimate the arameters of the corresonding model in the revious ste. The arameter estimation is based on maximizing a likelihood function for the available data. A conditional likelihood function is selected in order to get a good starting oint to obtain an exact likelihood function, as described in []. 2.4 Ste 3 In this ste, a diagnosis check is used to validate the model assumtions of Ste. This diagnosis checks if the hyotheses made on the residuals (actual rices minus redicted rices by estimated model in Ste 1 are true. Residuals must satisfy the requirements of a white noise rocess: zero mean, constant variance, uncorrelated rocess and normal distribution. These requirements can be checked by taking tests for randomness, such as the one based on the Ljung-Box statistic, and observing lots, such the autocorrelation and artial autocorrelations lots. If the hyotheses on the residuals are validated by tests and lots, then, the corresonding model can be used to forecast rices. Otherwise, the residuals contain a certain structure that should be studied to refine the model in Ste 1. This study is based on a careful insection of the autocorrelation and artial autocorrelation lots of the residuals. 2. Ste 4 In Ste 4, the corresonding model from Ste 2 can be used to redict future values of rices (tyically 24 hours ahead. Due to this requirement, difficulties may arise because redictions can be less certain as the forecast lead time becomes larger. The SCA System is again used to comute the 24- hour forecast. The exact likelihood function is selected to obtain a very accurate rediction. As a result of these five stes, the final models for the Sanish and Californian electricity markets are the following: 2..1 Dynamic regression Final selected arameters ω l of function ω (B in (1 that are different from zero are those corresonding to indices l = 1, 2, 3, 24,, 48, 49, 72, 73, 96, 97, 1, 121, 144, 14, 168, 169, 192, Transfer function Final selected arameters different from zero for d function ω (B in (2 are those corresonding to indices l =, 1, 2, 3, 24,, 48, 49, 72, 73, 96, 97, 1, 121, 144, 14, 168, 169, 192, 193. Selected arameters φ l of function φ (B in (2 that are different from zero are those corresonding to indices l = 1, 2, 3, 24,, 48, 49, 72, 73, 96, 97, 1, 121, 144, 14, 168, 169, 192, 193. Due to seasonality, function θ (B has been divided into two functions θ ( B = θ1( B θ 2 ( B where θ Θ 1 ( B l = 1 l θ =1 1 l B and θ = Θ 2 ( B l 1 l θ =1 2 l B. Selected arameters θ 1l for the first factor are different from zero for indices l = 1, 2, 3, 24. The only arameter different from zero for the second factor is at index l = ARIMA Final models for the Sanish and Californian electricity markets are, resectively:

4 ( 1 φ1b φ2 B φ3b φ4 B φb ( 1 φ23b φ24b φ47 B φ48b φ72b φ96b φ1 B φ144b (1 φ168b φ336 B φ4b lo g t = c + ( 1 θ1b θ 2 B ( 1 θ 24 B (1 θ168b θ 336B θ 4B εt ( 1 φ1b φ2b (1 φ23b φ24b φ47 B φ48b φ72b φ96b φ1b φ144b ( 1 φ167b φ168b φ169b φ192b (1 B(1 B (1 B log t = c + ( 1 θ1b θ2b ( 1 θ24b θ48b θ72b θ96b (1 θ144b (1 θ168b θ336b θ4b ε t Note that, as mentioned in Ste, the roosed formulation extends the standard ARIMA model by including more than two factors in (6 and (7, and a secial olynomial structure of the overall function. It should be noted that model (6 needs the revious hours to redict the next hour, whereas (7 just needs the revious two hours. Also, the model in (6 does not use differentiation, and the one in (7 uses hourly, daily and weekly differentiations: (1 B(1 B (1 B. This is related to the stationarity roerty of the series, and it can be traced by insecting the autocorrelation and artial autocorrelation lots. 3 NUMERICAL RESULTS (6 (7 3.1 Case Studies The three models described in the revious section have been alied to redict the electricity rices of mainland Sain and Californian markets. For the Sanish electricity market two weeks have been selected to forecast and validate the erformance for each of the three models. The first week corresonds to the second week of May 1 (from days May 11 th to 17 th, which is tyically a high demand week. The second one corresonds to the fourth week of August 1 (from th to 31 st which is a tyically a low demand week. The hourly data used to forecast the first week are from January 1 st to May th, 1; and the hourly data used to forecast the second week are from June 1 st to August 24 th, 1. For the Californian electricity market, the week of Aril 3 rd to 9 th has been chosen. This week is rior in time to the beginning of the dramatic rice volatility eriod. The hourly data used to forecast this week are from January 1 st to Aril 2 nd,. The ARIMA models (6-(7, corresonding to the Sanish and Californian markets, are used in the three roosed study cases. The dynamic regression and transfer function models are unique for both markets and all the case studies. 3.2 Numerical Results Numerical results for the three roosed models are resented. Fig. 1 to 9 show the forecasted rices for each of the three models and for each of the three weeks studied, two for the Sanish electricity market and one for the Californian market, together with the actual rices. Figure 1 corresonds to the selected week of May using a dynamic regression model for the Sanish market. Figure 1: of May week using a dynamic regression model in the Sanish market. Prices in /MWh. Table 1. The daily mean errors are around %. It can be observed that the hours with higher rediction errors are those corresonding to weekend days (Saturday and Sunday. Mean(% Table 1: Daily mean errors of May week using a dynamic regression model in the Sanish market. Figure 2 corresonds to the selected week of May using a transfer function model for the Sanish market May Week (DR May Week (TF Figure 2: of May week using a transfer function model in the Sanish market. Prices in /MWh.

5 The seven daily mean errors for this week are shown in Table 2. The daily mean errors are around.2%. On Sunday the error is greater. Mean(% Table 2: Daily mean errors of May week using a transfer function model in the Sanish market. Figure 3 corresonds to the same week in May using the ARIMA model (6 for the Sanish market. Figure 3: of May week using an ARIMA model in the Sanish market. Prices in /MWh. Table 3. The daily mean errors are around 8%. Mean(% Table 3: Daily mean errors of May week using an ARIMA model in the Sanish market. Figure 4 corresonds to the selected week of August using a dynamic regression model for the Sanish market May Week (AM August Week (DR Figure 4: of August week using a dynamic regression model in the Sanish market. Prices in /MWh. Table 4. The daily mean errors are around 4.%. Mean(% Table 4: Daily mean errors of August week using a dynamic regression model in the Sanish market. Figure corresonds to the same week of August using a transfer function model for the Sanish market August Week (TF Figure : of August week using a transfer function model in the Sanish market. Prices in /MWh. Table. The daily mean errors are around 4.4%. A good erformance of the rediction method can be observed. Mean(% Table : Daily mean errors of August week using a transfer function model in the Sanish market. Figure 6 corresonds to the same week of August using the ARIMA model (6 for the Sanish market August Week (AM Figure 6: of August week using an ARIMA model in the Sanish market. Prices in /MWh. Table 6. The daily mean errors are around 7.8%.

6 Mean(% Table 6: Daily mean errors of August week using an ARIMA model in the Sanish market. Figure 7 corresonds to the selected week in Aril using a dynamic regression model for the Californian market Aril Week (DR Figure 7: of Aril week using a dynamic regression model in the Californian market. Prices in \$/MWh. Table 7. The daily mean errors are around 3.3%. Mean(% Table 7: Daily mean errors of Aril week using a dynamic regression model in the Californian market. Figure 8 corresonds to the same week in Aril using a transfer function model for the Californian market Aril Week (TF Figure 8: of Aril week using a transfer function model in the Californian market. Prices in \$/MWh. Table 8. The daily mean errors are around 3.%. Mean(% Table 8: Daily mean errors of Aril week using a transfer function model in the Californian market. Figure 9 corresonds to the same week in Aril using the ARIMA model (7 for the Californian market. Prices (\$/MWh Aril Week (AM Figure 9: of Aril week using an ARIMA model in the Californian market. Prices in \$/MWh. Table 9. The daily mean errors are around %. During the weekend the errors are greater. Mean(% Table 9: Daily mean errors of Aril week using an ARIMA model in the Californian market. Several statistical measures have been used to verify the rediction ability of the roosed models. For all the study cases, the average rediction error of the 24 hours has been comuted for each day. Then, the average of the daily mean errors has been calculated and called Mean Week Error (MWE. Finally, the ed Mean Square Error (FMSE for the 168 hours of each week has been calculated as: FMSE = ( t ˆ t (8 i= 1 where t and ˆt are the actual and forecasted rices, resectively. An index of uncertainty in the models is the variability of what is still unexlained after fitting the models. That can be measured through the variance of the error term (σ 2. The smaller σ 2 the more recise the rediction of rices. Normally, the value of σ is not known, thus an estimate is used instead. The standard deviation of the error terms, ŝ R, can be used as such an estimate. This estimate is useful when the true values of the series are unknown.

7 These measures can be observed in Table. First column indicates the month, and in arenthesis the model used: DR for Dynamic Regression, TF for Transfer Function and AM for ARIMA model. The second column shows the ercentage Mean Week Error (MWE, the third one resents the standard deviation of the error terms ( ŝ R, and the fourth column shows the square root of the ed Mean Square Error (FMSE. Note that rices, ŝ R, and FMSE are measured in /MWh and \$/MWh in the Sanish and Californian markets, resectively. MWE (% ŝ R FMSE Sain May 1 (DR Sain May 1(TF Sain May 1 (AM Sain August 1 (DR Sain August 1(TF Sain August 1(AM California Aril (DR California Aril (TF California Aril (AM Table : Statistical Measures. Finally, Tables 11 and 12 show the MWE for the last week of the first ten months of the year in Sain, and the same week for the months of August and November in California. After Aril the Californian market entered a highly unstable eriod that later rovoked the collase at the end of the year. Exlanatory variables such as: demand, water storage and the available roduction of hydro units, are considered in the ARIMA case. MWE (% DR TF ARIMA ARIMA-EX 1 January February March Aril May June July August Setember October Table 11: Mean Week Error for the last week of the first ten months of in the Sanish Market. MWE (% DR TF ARIMA ARIMA-EX 2 August November Table 12: Mean Week Error for the last week of August and November, in the Californian Market. 1 With exlanatory variables: demand, water storage and available hydro roduction. 2 With exlanatory variable: demand. All the study cases have been run on a DELL Precision 6 Workstation with two rocessors Pentium III, 1 Gb of RAM, and 8 MHz. Running time, including estimation and forecasting, has been under three minutes in all cases. 4 CONCLUSIONS This aer has roosed three forecasting models: dynamic regression, transfer function and ARIMA, to redict hourly electricity rices in the Sanish and Californian day-ahead markets. These models are based on time series analysis. For all markets and case studies, the dynamic regression and transfer function models have erformed better than the ARIMA model, though the three techniques rovide reasonable redictions. The difference between the ARIMA model and the other two may be due to the lack of flexibility of the ARIMA formulation when including multile seasonality terms. The effect of choosing a low versus a high demand week has been negligible to the forecasts. However, the effect of the exlanatory variables has imroved the ARIMA redictions, albeit not always. The forecasted rices have shown better behavior in the Californian market before the crash that took lace in the summer of. This could be due to the fact that that market showed less volatility and a lower roortion of outliers before the turmoil. The Sanish model has needed hours to redict future rices, instead of the 2 hours needed in the Californian one. These differences may reflect different bidding behaviors and that will be subject of future research. ACKNOWLEDGMENT This work was artially suorted by the Ministry of Science and Technology (Sain and the Euroean Union through grant FEDER-CICYT 1FD We are also grateful to Mr. Bill Lattyak, from SCA, for his hel and advice. REFERENCES [1] J. M. Arroyo and A. J. Conejo, Otimal Resonse of a Thermal Unit to an Electricity Sot Market, IEEE Transactions on Power Systems, vol., no. 3, , August. [2] A. J. Conejo, J. Contreras, J. M. Arroyo and S. de la Torre, Otimal Resonse of an Oligoolistic Generating Comany to a Cometitive Pool-Based Electric Power Market, to aear in IEEE Transactions on Power Systems, aer TR2. [3] G. Gross and F. D. Galiana, Short-Term Load ing, Proc. of the IEEE, vol. 7, no. 12,. 8-73, December [4] O. B. Fosso, A. Gjelsvik, A. Haugstad, M. Birger and I. Wangensteen, Generation Scheduling in a Deregulated System, IEEE Transactions on Power Systems, vol. 14, no. 1,. 7-81, February 1999.

8 [] B. Ramsay and A. J. Wang, A Neural Network Based Estimator for Electricity Sot-Pricing with Particular Reference to Weekend and Public Holidays, Neurocomuting, vol. 23,. 47-7, [6] B. R. Szkuta, L. A. Sanabria and T. S. Dillon, Electricity Price Short-Term ing using Artificial Neural Networks, IEEE Transactions on Power Systems, vol. 14, no. 3, , August [7] J. D. Nicolaisen, C. W. Richter, Jr. and G. B. Sheblé, Signal Analysis for Cometitive Electric Generation Comanies, Proc. of the Conference on Electric Utility Deregulation and Restructuring, London, United Kingdom, 4-7 Aril,. [8] J. J. González and P. Basagoiti, Sanish Power Exchange and Information System Design Concets, and Oerating Exerience, Proc. of the 21 st Power Industry Comuter Alications International Conference, PICA 99, Santa Clara, USA, May [9] F. Albuyeh and Z. Alaywan, Imlementation of the California Indeendent System Oerator, Proc. of the 21 st Power Industry Comuter Alications International Conference, PICA 99, Santa Clara, USA, May [] G. E. P. Box and G. M. Jenkins, Time Series Analysis ing and Control, 2 nd ed., Holden-Day, San Francisco, USA, [11] F. J. Nogales, J. Contreras, A. J. Conejo and R. Esínola, ing Next-Day Electricity Prices by Time Series Models, to aear in IEEE Transactions on Power Systems, aer 1TR212. [12] L. Liu and G. P. Hudak, ing and Time Series Analysis using the SCA Statistical System, Scientific Comuting Associated, 1994.

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