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This paper considers forecasting techniques to predict the 24 market-clearing prices of a day-ahead electric energy market. The techniques considered include time series analysis, neural networks and wavelets. Within the time series procedures, the techniques considered comprise ARIMA, dynamic regression and transfer function. Extensive analysis is conducted using data from the PJM Interconnection. Relevant conclusions are drawn on the effectiveness and flexibility of any one of the considered techniques. Furthermore, they are exhaustively compared among themselves.

1.1. Electric energy markets During the last two decades the electric power industry all over the world has undertaken significant restructuring. In most countries, a cost minimization paradigm has been replaced by a profit maximization one. In the cost minimization framework, a Central Operator (CO) decides centrally how the generating machines should be operated to minimize total cost while serving all demands. In contrast, in the profit maximization framework, producers, retailers and consumers interact through a market seeking to maximize their respective profits. Two market structures arise commonly in practice: a bilateral contract framework and a pool. In a bilateral transaction market, any given producer agrees with retailers to supply specified amounts of energy during a contract horizon. Those physical contracts are implemented with the help of an Independent System Operator (ISO) that takes care of the physical requirements needed for the transactions to take place in a secure manner. In a pool, producers submit to the Market Operator (MO) production bids that typically consist of a set of energy blocks and their corresponding minimum selling prices for every hour of the market horizon. Analogously, retailers and large consumers submit to the MO consumption bids that consist of a set of energy blocks and their corresponding maximum buying prices. The MO uses a market-clearing algorithm to clear the market, which results in a market-clearing price as well as the scheduled production and consumption for every hour of the market horizon. The market-clearing price is the price to be paid by retailers and to be charged by producers. Finally, the ISO checks for technical feasibility and, if needed, introduces the minimal required changes to attain a secure operation. This pool-based electric energy market is the most common arrangement in practice. Sometimes, it coexists with a bilateral contract framework. In this case, the ISO ensures a technical secure operation of the pool plus the simultaneous bilateral contract arrangements. This paper considers the framework above, i.e., a pool that may include bilateral contract arrangements. 1.2. Why forecasting electricity prices? Producers need to forecast market-clearing prices to respond optimally to the pool and to efficaciously engage in bilateral contracts. In the short-run, a producer with low capability of altering market-clearing prices (price-taker producer) needs day-ahead price forecasts to optimally self-schedule and to derive its bidding strategy in the pool. In the medium-term, a price-taker producer requires market-clearing price forecasts for several months in order to sign favorable bilateral contracts. Retailers and large consumers need day-ahead and medium-term market-clearing price estimates for the same reasons as producers. Those price forecasts constitute fundamental information for the retailers (large consumers) to self-schedule and to bid efficiently in the pool; and to engage in profitable bilateral contracts. 1.3. Forecasting framework Within the framework of a pool-based electric energy market, this paper considers forecasting techniques to estimate the 24 day-ahead market-clearing prices. Three families of techniques are considered: time series, neural networks and wavelets. Time series techniques are treated with greater detail because they revealed themselves, through many realistic studies, as the most efficacious tools for day-ahead market-clearing price forecasting. Time series techniques considered include ARIMA, dynamic regression and transfer function. A naïve but challenging test is used to characterize all forecasting procedures that are analyzed. The 24 market-clearing price forecasts using any technique can be compared to the 24 market-clearing prices of a day similar to the one to be forecast. A similar day is characterized as follows. A Monday is similar to the Monday of the previous week and the same rule applies for Saturdays and Sundays; analogously, a Tuesday is similar to the Monday of the same week, and the same rule applies for Wednesdays, Thursdays and Fridays. The naïve test is passed if hourly errors for the estimates using any forecasting technique are smaller than the market-clearing prices of the similar day. More often than expected, forecasting procedures not carefully tuned up do not pass this test. This naïve test is used in the case studies that are analyzed in this paper. The time framework to forecast day-ahead market-clearing prices in most markets is explained below and illustrated in Fig. 1. The market-clearing price forecasts for day d are required on day d−1, typically at hour hb (around 10 am). On the other hand, data concerning results for day d−1, including market-clearing prices and demands, are available on day d−2 at hour hc (around noon). Therefore, the actual forecasting of market-clearing prices for day d can take place between hour hc of day d−2 and hour hb of day d−1. Additionally, producers and retailers use this period to optimally self-schedule and to produce appropriate bids. To perform a fair comparison, available data for price prediction is identical for all techniques. To compute price forecasts for hour 1 to 24 of day d, data available to all procedures include price and demand historical data up to hour 24 of day d−1, and demand predictions for the 24th hour of day d. The considered historical data spans 53 days. However, this available data is not used in the same way by all techniques. In the case of ARIMA and neural network models, historical demand data does not significantly improve predictions and therefore it was decided that these models would only rely on price data. In all techniques the corresponding algorithm uses endogenously computed price forecasts of some hours of day d to forecast prices of future hours of the same day d. For instance, using an ARIMA model to forecast the price of hour 5 of day d, historical price data up to hour 24 of day d−1 are used, as well as previously obtained price forecasts for hour 4, 3 and 2 of day d. The only explicative variable that has been considered in the study is the demand. The amount of hydro energy stored in reservoirs is a variable of interest for electric energy markets with important hydroelectric producers. However, in this study a market that includes only thermal producers is considered. Explicative variable data (temperature, hydro resources, thermal equipment outages and the like) for day d, if available, can be easily and fruitfully included in the models presented. Note, however, that unit outage information is generally proprietary and therefore not available to all market agents. Note also that the effect of the temperature, and other weather-related variables, is usually embodied in the demand forecasts. Nevertheless, for the sake of simplicity and clarity in the comparisons, explicative variables other than demand have not been included in the models presented. 1.4. Time series characterization In most electricity markets the series of prices presents the following features: 1. High frequency, 2. Non-constant mean and variance, 3. Daily and weekly seasonality, 4. Calendar effect on weekend and holidays, 5. High volatility, 6. Presence of outliers. The presented models are selected once these characteristics are inspected and understood. These characteristics can be observed in Fig. 2 for the price series of the PJM Interconnection (PJM, 2003) for the year 2002. In this figure, the actual hourly price is the demand-weighted average of all day-ahead locational marginal prices in that hour for the whole PJM interconnection. These prices are available from PJM (2003). The demand for electricity partly explains price behavior. Fig. 3 depicts hourly electricity demand for the year 2002 in the PJM Interconnection. It should be noted that as a result of the possibly irrational bidding behavior by market agents (producers and consumers), price series are more volatile than demand series. The non-constant mean feature of the price series is alleviated by differentiating the original series using factors of the form (1−Bs), where B is the back-shift operator and s can be 1 (hourly differentiation), 24 (daily differentiation), 168 (weekly differentiation), or other values depending on the series. The non-constant variance is alleviated by taking logarithms. Fig. 4 shows the resulting differentiated logarithmic price series. It can be observed that this series has relatively constant mean and variance. Daily and weekly seasonalities are typically taken into account through the use of seasonality models of orders 24 and 168, respectively. The calendar effect is taken into account by incorporating ad-hoc logic. The high frequency and high volatility features are characteristics inherent to the series that cannot be changed. Outliers have not been explicitly treated. However, the automatic handling of outliers provided by the estimation software used (SCA, 2003) did not improve predictions in the case studies analyzed. 1.5. Overview of forecasting procedures The three time series models presented constitute a class of stochastic processes used to analyze time series, and they share a common methodology due to Box, Jenkins, and Reinsel (1994). The analysis is based upon different probability models to represent the data, including the explanatory variables. The three techniques differ when modeling the relationship between prices and errors. ARIMA relates current prices to past prices, and current errors to previous errors; dynamic regression relates current and past prices and demands; and transfer function relates current prices to past prices, demands and errors. Artificial Neural Networks (ANNs) are mathematical models that resemble the functioning of the human brain. ANNs need to be trained with a set of inputs to produce the specified outputs. They are composed of an input layer, one or more hidden layers and an output layer. When forecasting time series, ANNs are interesting because they can approximate any nonlinear function. In particular, feedforward backpropagation neural networks are specially suited to forecasting electricity prices because they can process nonlinearities using sigmoid functions for the inputs and linear functions for the outputs. Wavelet Transform Analysis (WTA) has been extensively used for various signal processing applications in the past, but its use in predicting electricity prices is recent. It is an alternative to the classical Fourier Transform (FT) for analyzing a non-stationary series. The basic difference between the FT and WTA is that WTA uses short windows at high frequencies and long windows at low frequencies, in contrast to FT, which uses a single window. 1.6. Literature review and paper contributions Regarding time series analysis, the following references are relevant: dynamic regression and transfer function papers include Nogales, Contreras, Conejo, and Espínola (2002); ARMA papers include Fosso, Gjelsvik, Haugstad, Birger, and Wangensteen (1999); and ARIMA papers include Contreras, Espínola, Nogales, and Conejo (2003) in the market of mainland Spain. Artificial intelligence techniques, such as ANNs, have been applied to forecasting prices in the England–Wales pool (Ramsay & Wang, 1998), the Australian market (Szkuta, Sanabria, & Dillon, 1999), the PJM Interconnection (Hong & Hsiao, 2002), and the New England ISO, (Zhang, Luh, & Kasiviswanathan, 2003). In addition, wavelet transform analysis has been used in the England–Wales pool (Kim, Yu, & Song, 2002). Relevant references not directly related to electricity price forecasting follow. References on gas and oil price forecasting include Morana (2001), Cabedo and Moya (2003) and Buchananan, Hodges, and Theis (2001). Electricity demand is forecast in Darbellay and Slama (2000) and Harris and Liu (1993). Residential gas consumption forecasting is analyzed in Liu and Lin (1991) and forecasting in oil markets in treated in Abramson and Finizza (1995). The main contributions of this paper are: 1. Time series algorithms, including dynamic regression, transfer function and ARIMA are developed to forecast day-ahead electricity prices. The building of these models is based on a thorough analysis of historical price and demand data. 2. Two unconventional techniques, a neural network algorithm and a wavelet transform procedure, are proposed to predict day-ahead prices. The neural network procedure relies on precise optimization algorithms. 3. All prediction techniques are compared using a realistic case study based on electricity price data from the PJM Interconnection corresponding to year 2002. Appropriate conclusions are drawn. 1.7. Paper organization This paper is organized as follows. In Section 2, the three families of forecasting procedures are described. As regards time series analysis, the techniques considered include ARIMA, dynamic regression and transfer function. Additionally, procedures based on neural networks and wavelets are analyzed. In Section 3, a case study based on data from the PJM Interconnection for the year 2002 is analyzed in detail. Section 4 provides some relevant conclusions.

This paper analyzes different forecasting techniques to predict the 24 market-clearing prices of a day-ahead electric energy market. Price forecasting, both in the short- and the long-term, is required by producers, retailers and consumers to determine their respective bidding strategies in the pool and to engage in beneficial bilateral contracts. Extensive testing has been carried out using data from the PJM Interconnection in the year 2002. These data correspond to day-ahead prices, which are demand-weighted averages of locational marginal prices. Time series techniques reveal themselves as more efficacious than wavelet-transform or neural network techniques. One reason for this is that both wavelet and neural network techniques break the time series of market-clearing prices in some manner. Among time series techniques, the dynamic regression and transfer function algorithms are more effective than ARIMA models. Wavelet models behave similarly to ARIMA models and neural network procedures do not show good performance. It is remarkable the reasonable results obtained in some cases using the naïve technique. A promising idea for further research is the combination of wavelet transform and time series algorithms. That is, the historical price series is decomposed using the wavelet transform; the resulting detail and approximation series, which are well behaved, are individually used to forecast their respective future behavior using transfer function algorithms. Then these forecast series (detail and approximation) are used to reconstruct a price forecast series using the inverse wavelet transform algorithm.