Electrical Power and Energy Systems 33 (2011) 550–555
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Electricity price forecasting using artiﬁcial neural networks Deepak Singhal, K.S. Swarup ⇑ Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
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Article history: Received 5 December 2006 Received in revised form 1 December 2010 Accepted 9 December 2010 Available online 1 February 2011 Keywords: Forecasting Artiﬁcial neural networks Open power market Power trading Market-clearing price (MCP) Price forecasting
a b s t r a c t Electricity price forecasting in deregulated open power markets using neural networks is presented. Forecasting electricity price is a challenging task for on-line trading and e-commerce. Bidding competition is one of the main transaction approaches after deregulation. Forecasting the hourly market-clearing prices (MCP) in daily power markets is the most essential task and basis for any decision making in order to maximize the beneﬁts. Artiﬁcial neural networks are found to be most suitable tool as they can map the complex interdependencies between electricity price, historical load and other factors. The neural network approach is used to predict the market behaviors based on the historical prices, quantities and other information to forecast the future prices and quantities. The basic idea is to use history and other estimated factors in the future to ‘‘ﬁt’’ and ‘‘extrapolate’’ the prices and quantities. A neural network method to forecast the market-clearing prices (MCPs) for day-ahead energy markets is developed. The structure of the neural network is a three-layer back propagation (BP) network. The price forecasting results using the neural network model shows that the electricity price in the deregulated markets is dependent strongly on the trend in load demand and clearing price. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Market operations in electric power systems involve the determination of forecasted values of electricity prices in addition to the load demand over a future horizon. Independent Power Producers (IPPs) or market players depend on the forecasted values of electric prices to decide strategies to broadcast ‘sell’ and ‘buy’ bids for selling and buying of power in the power trading market. Spot pricing of electricity requires the determination of the electricity price in real-time. Accurate forecasting of electricity prices are necessary for the entities to participate in the biding process. Knowledge of the electricity prices over a wider horizon are required for day ahead market in deciding the units to be committed, termed as price based unit commitment, to bidding available power generation over the operating scenario. The electric power industry has over the years been dominated by large utilities that had an authority over all activities in generation, transmission and distribution of power within its domain of operation. Such utilities have often been referred to as vertically integrated utilities. Such utilities served as the only electricity provider in the region and were obliged to provide electricity to everyone in the region. The utilities being vertically integrated, it was often difﬁcult to segregate the costs incurred in generation, transmission or distribution. Therefore, the utilities often charged their customers an average tariff rate depending on their aggregated ⇑ Corresponding author. Tel.: +91 44 2257 4440; fax: +91 44 2257 4402. E-mail address: [email protected]
(K.S. Swarup). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.12.009
cost during a period. The price setting was done by an external regulatory agency and often involved considerations other than economics. The wholesale power markets have been growing everywhere at a fast pace because of the ongoing deregulation of the power industry. Power production industry is clearly demarked from power transmission industry. Power trading as a result has become a very important part of power industry. An important input to the decision-making activities of a Genco is a good forecast of the market prices. This is important because an accurate forecast of the short-term market price helps the Genco to bid for power sell or buy appropriately and strategically, thereby providing higher returns. Bilateral contract prices also have a tendency to be indirectly affected by spot-price trends. Thus good spot market price forecasts can help set up proﬁtable bilateral contracts. In the short-term markets, continuous trading up to 2 h in advance of real-time is possible. In these markets, the prices can be highly volatile to system conditions such as sudden outages, and external factors such as temperature variations, and rainfall. It is usually of great interest to Gencos and other market players to have a good forecast toolbox for these prices. Price forecast in the general sense also include forecast of futures and forward market prices . These forecasts may be carried out months or even a year in advance. These forecasts may be useful if the Genco is contemplating investments in generation capacity, market risk analysis, production and maintenance planning, among others. Most often the Genco has an in-house price forecast tool based on available forecasting methods such as the conventional linear regression analysis technique, to cater to the need of a price forecast.
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1.1. Electricity pricing In a power market, the price of electricity is the most important signal to all market participants and the most basic pricing concept is market-clearing price (MCP). Generally, when there is no transmission congestion, MCP is the only price for the entire system. However, when there is congestion, the zonal marketclearing price (ZMCP) or the Locational Marginal Price (LMP) could be employed. ZMCP may be different for various zones, but it is the same within a zone. LMP can be different for different buses. The bidding decision making process for optimal electricity supply is formulated as a Markov decision process. The suppliers are modeled with their bidding parameters with corresponding transition probabilities. Fundamental conceptual framework for market and bidding decision making is presented in . Market clearing tool for market operator of a pool based electricity market is presented. The problem is formulated as a mixed-integer linear programming problem. The results of the new market clearing procedure are presented for 20 generating units for 24 h duration . The most distinct property of electricity is its volatility. Volatility is the measure of change in the price of electricity over a given period of time. It is often expressed as a percentage and computed as the annualized standard deviation of percentage change in the daily price (other prices such as weekly or monthly prices can also be used), Compared with load, the price of electricity in a restructured power market is much more volatile. From the curves, we learn that the load curve is relatively homogeneous and its variations are cyclic and the price curve is nonhomogeneous and its variations show a little cyclic property. Although electricity price is very volatile, it is not regarded as random. Hence, it is possible to identify certain patterns and rules pertaining to market volatility. For example, transmission congestion usually incurs a price spike which is not sustained as electricity price would revert to a more reasonable level (this is known as mean reversion in statistics). It is conceivable to use historical prices to forecast electricity prices. Accordingly, we use a training scheme to capture perceived patterns for forecasting electricity prices. The fundamental reason for electricity price spike is that the supply and demand must be matched on a second-by-second basis. Other reasons follow:
Volatility in fuel price Load uncertainty Fluctuations in hydroelectricity production Generation uncertainty (outages) Transmission congestion Behavior of market participant (based on anticipated price) Market manipulation (market power, counterparty risk)
Because of the special properties of electricity, the price of electricity is far more volatile than that of other relatively volatile commodities. The annualized volatility of oil future contracts is around 30%; it is around 50% for natural gas future contracts, while about 60% for electricity future contracts. In electricity spot markets, annualized volatility is above 200%. Because of the signiﬁcant volatility, it is difﬁcult to make an accurate forecast for the spot market of electricity. This is evidenced by the fact that the existing price forecasting accuracy is far lower than that of load forecasting. However, price forecasting accuracy is not as stringent as that of load forecasting. The power awarded to each bidder is determined based on the individual bid curves and the MCP. All the power awards will be compensated at the MCP. After the auction closes, each bidder aggregates all its power awards as its system demand, and per-
forms a traditional unit commitment or hydrothermal scheduling to meet its obligations at minimum cost over the bidding horizon. Suppliers bidding decisions are coupled with generation scheduling since generator characteristics and how they will be used to meet the accepted bids in the future have to be considered before bids are submitted. Therefore bidding decision must consider the anticipated MCP, generation award and costs, and competitor’s decisions. The MCP and MCQ (Market Clearing Quantity) are the most important power market indicators. Forecasting the hourly MCP and MCQ in daily power markets is the most essential task and basis for any decision making in the power market. 1.2. Electricity price forecasting methods There are various methods adopted for the forecasting of future market price. One approach to predict the market behaviors is regression. The basic idea is to use the historical prices, quantity and other information such as load forecast, and temperatures to predict the MCPs. That is, use history and other estimated factors in the future to ‘‘ﬁt’’ and ‘‘extrapolate’’ the prices and quantity. Important methodological issues and techniques for electricity load and price forecasting are presented in . Computationally intensive methods like variable segmentation, multiple modeling, combinations and neural networks for forecasting demand side and strategic simulation using artiﬁcial agents for the supply side are used. Conceptual framework for designing price forecasting approaches are presented . Modeling competitive market behavior in capturing uncertainty in inputs/outputs with adaptability and transparency is presented. Model of Market-clearing price (MCP) and Database for forecasting electricity prices is described. Forecasting Energy prices using neural networks and fuzzy logic and their combination is discussed in . Historical behaviors of spot prices was evaluated for these methods. Emphasis is placed on the identiﬁcation of important parameters which inﬂuence the forecasted quantity. Basic framework of artiﬁcial neural network for load forecasting based on historical load data and temperature is presented . A multi layer perception using three hidden layers is implemented for accurate load forecasting. A phase space is reconstructed from the scalar time series representing the chaotic characteristics of electricity price. The main features of thee attractors are extracted and the surrogate data method is used. Global and Local price forecasting model based recurrent neural network is proposed and applied for the New England Market. . An artiﬁcial intelligence method using both fuzzy C- means (FCM) algorithm and recurrent neural network (RNN) is used for forecasting the LMPs. The RNN were trained using historical prices for two months from Pennsylvania, New Jersey and Maryland (PJM). The RNN was found to forecast electricity prices with reasonable amount of accuracy . System marginal price short term forecasting (48 h) using a three layer artiﬁcial neural network employing data from Victorian power system is presented . Model sensitivity test for input variable selection validation is discussed to justify the concept of inﬂuence of input variables on the output. Electricity price forecasting based on chaos theory is presented which is based on the fact that electricity price possesses chaotic characteristics, where the Lyapunov and components and fractal dimensions of the attractors are extracted. An accurate phase space is reconstructed by multivariable time series constituted by electricity price and its correlated factors. A recurrent neural network is employed for Global and Local electricity price forecasting [10,11].
D. Singhal, K.S. Swarup / Electrical Power and Energy Systems 33 (2011) 550–555
The inputs to the Neural Network forecaster are:
2. Problem description and formulation of proposed methodology 2.1. Factors considered in price forecasting There are many factors that may inﬂuence the market auction result, such as system load of the entire area covered by the market, power import to and export from outside the market through long term contract, the available hydro energy, fuel price, etc. In order to forecast MCP, these factors can be used as input variables if available. Therefore input variable selection is extremely important. Based on experience of the market analysts and correlation analysis, the following factors can be considered as input variables. 2.1.1. Historical MCPs The historical MCPs are natural selections since history and future are correlated. The hourly MCPs demonstrate some cyclic characteristics. The basic cycle is 24 h. In a week span, each day’s pattern would be different especially between a weekday and weekend. Therefore week is also a cycle. On the other hand, the system load is different in a year for different seasonal climate. This would be reﬂected in the MCPs. Clearly year is a cycle too. Therefore there are at least three cycles in MCPs: day, week and year. The average on peak and off peak MCPs of last few weeks as inputs can provide the trend information over the recent past. 2.1.2. System loads Load ﬂuctuations could impact price. On the other hand, price ﬂuctuations could impact load values. Thus, load forecasting and price forecasting can be combined into a single forecasting model. Therefore the historical load and forecasted load are used as input. 2.1.3. Fuel prices The fuel costs are main part of total generation cost. The change in fuel prices may affect the market prices. A well-established nonlinear regression method is artiﬁcial neural network. Neural networks have been used for pool price forecasting . Several techniques based on neural networks, fuzzy systems and other intelligent methods have been widely employed for forecasting the electricity price [13,14].
1. 2. 3. 4. 5. 6. 7. 8. 9.
Day of week Time slot of Day Forecasted Demand i.e. D(t) Change in demand i.e. D(t) D(t-1) Price (one day ago) – 3 inputs i.e. P(t-47), P(t-48), P(t-49) Price (one week ago) – 3 inputs i.e. P(t-335), P(t-336), P(t-337) Price (two weeks ago) – 1 input i.e. P(t-672) Price (three weeks ago) – 1 input i.e. P(t-1008) Price (four weeks ago) – 1 input i.e. P(t-1344)
Table 1 shows the inputs to the neural network for electricity price forecasting. The ﬁrst two inputs are chosen as they are the time indices. A third and fourth input represents the current status of the market and as demand is inter related with price, they are one of the
Table 1 Neural network input for electricity price forecasting. No. 1 2 3 4 5 6 7 8 9
Time information Load demand Historical price information
Day of week Time slot of day Forecasted demand Change in demand Price (one day ago) – 3 inputs Price (one week ago) – 3 inputs Price (two weeks ago) – 1 input Price (three weeks ago) – 1 input Price (four weeks ago) – 1 input
1 1 1 1 3
t D(t) D(t) D(t-1) P(t-47),P(t48),P(t-49) P(t-335),P(t336),P(t-337) P(t-672)
2.2. Implementation of the forecasting model The electricity price data was collected for eight months. The Neural Network is trained with the data of six months. It is tested for various days of a particular month. The days include the day with normal trend, day with small spike and the day with large spike. The collected data contains eight months data which gives the values of total demand and price at every time slot. The time steps are half hourly which mean that one day contains 48 time steps. In our model we will take the historical prices, historical and forecasted demands and time indices as inputs as these are the information that is gathered from the data. The inputs to the neural network consist of the time indices, electricity price and load demand data. Historical information of electricity prices and past load demand constitutes important inputs for predicting the electricity price. The output of electricity price can take several durations, namely hourly, daily and weekly forecasting. Forecasting of load demand is mostly hourly, whereas electricity price is for more frequently to facilitate spot pricing of electricity. Power trading in electric power markets usually use price signals varying over a wide range. Day ahead markets use requires forecasted prices at least 2 days in advance. To simulate the market power conditions, day ahead forecasting for electricity prices for 48 h is carried out in this work.
Fig. 1. Neural network model for price forecasting.
D. Singhal, K.S. Swarup / Electrical Power and Energy Systems 33 (2011) 550–555 Table 2 Predicted and actual values of prices for day with normal trend, small and large spike in prices. Time
Electricity price ($/MWh) for 48 h with normal trend, small and large spikes. I. Normal trend price
II. Price with small spike
III. Price with large spike
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 MAE RMSE
18.25 18.25 18.56 17.54 20.24 20.44 21.59 28.21 29.47 33.09 42.19 38.09 37.03 38.87 37.03 31.34 27.98 28.79 26.34 25.48 26.09 30.80 34.30 28.49 29.67 28.32 32.19 36.32 51.94 75.73 79.85 58.70 52.98 41.06 38.23 33.50 30.25 27.48 35.43 33.64 37.23 37.03 33.59 30.21 25.48 27.75 26.86 22.10 2.655 0.525
17.33 17.55 17.04 17.83 17.49 18.93 25.75 27.24 31.34 41.01 40.08 39.62 39.72 39.37 34.34 29.87 29.55 27.05 26.75 27.25 29.8 31.71 28.27 28.92 26.12 31.14 33.63 41.19 61.94 80.33 71.85 62.22 40.99 40.2 37.5 34.56 31.95 26.43 36.98 31.29 38.7 33.69 36.94 27.3 29.09 27.15 25.09 22.25
0.92 0.69 1.52 0.28 2.75 1.51 4.16 0.97 1.86 7.92 2.12 1.528 2.68 0.49 2.68 1.47 1.56 1.73 0.40 1.77 3.70 0.90 6.03 0.43 3.54 2.82 1.43 4.85 10.00 4.59 7.99 3.52 11.99 0.85 0.73 1.05 1.70 1.05 1.55 2.35 1.47 3.33 3.34 2.91 3.61 0.59 1.77 0.14
24.93 26.70 18.65 19.94 24.26 23.04 25.33 29.52 23.93 23.02 31.39 39.27 35.57 49.99 42.58 40.20 39.26 28.79 27.30 27.74 23.82 27.45 27.32 26.67 23.93 24.41 24.41 26.53 28.04 53.04 92.91 85.95 75.34 59.65 51.79 40.58 40.21 39.25 38.49 36.84 45.96 40.81 35.91 38.48 31.86 25.90 27.48 26.17 0.682 1.129
25.84 20.36 19.05 22.55 22.77 24.15 24.43 24.9 23.8 23.59 34.01 37.4 41.29 45.09 40.79 39.51 31.22 28.59 27.45 26.1 26.88 26.88 26.58 24.64 24.64 24.42 25.83 27.73 32.28 58.13 138.48 106.08 81.66 62.68 46.28 43.59 41.85 41.35 38.81 41.39 49.13 41.24 39.46 33.51 27.65 27.33 27.88 26.84
0.90 6.34 0.39 2.60 1.49 1.10 0.90 4.62 0.13 0.56 2.61 1.87 5.71 4.90 1.79 0.69 8.04 0.20 0.14 1.64 3.05 0.57 0.72 2.03 0.70 0.00 1.41 1.19 4.24 5.09 45.57 20.12 6.31 3.02 5.51 3.00 1.63 2.09 0.32 4.54 3.16 0.42 3.55 4.97 4.21 1.42 0.39 0.66
38.31 31.61 26.19 27.74 28.58 27.08 26.74 25.09 25.80 25.84 26.29 26.38 25.44 22.79 23.06 26.75 29.95 37.09 45.08 45.80 41.262 41.56 38.63 30.95 27.28 28.25 25.64 26.15 26.52 20.14 22.82 21.15 22.15 25.82 44.05 162.28 151.14 121.41 82.25 47.93 44.13 54.49 47.29 39.21 35.19 34.92 30.68 35.01 9.282 4.105
33.51 27.65 27.33 27.88 26.84 26.3 24.83 25.02 25.09 25.48 25.65 24.94 22.61 23.03 20.42 27.19 30.98 38.48 41.03 41.75 42.18 39.89 33.13 28.43 27.93 25.69 25.53 25.77 20.65 21.53 20.34 20.88 23.98 31.37 56.34 268.11 294.77 201.66 82.16 52.3 50.85 58.58 44.65 37.88 39.79 32.84 34.54 27.3
4.80 3.96 1.13 0.13 1.74 0.78 1.90 0.07 0.71 0.36 0.64 1.44 2.83 0.23 2.64 0.43 1.02 1.38 4.04 4.05 0.91 1.67 5.50 2.52 0.64 2.56 0.11 0.38 5.87 1.38 2.48 0.27 1.82 5.54 12.28 105.82 143.62 80.24 0.09 4.36 6.71 4.08 2.64 1.33 4.59 2.08 3.85 7.70
MAE: mean absolute error. RMSE: root mean square error.
important inputs. The rest of the inputs are historical price values. Fifth input represents the price one day ago at the same time step and the steps near it. This input represents the latest market trend as the day is a cycle. Sixth input represents the price one week ago at the same time step and the steps near it. This also represents the trend of the market for a longer period. The remaining inputs are the prices of two, three and four weeks ago respectively. These are considered since week is a cycle. Thus there are thirteen inputs used to forecast the system price at any given instant. Fig. 1 shows the Neural Network for price forecasting which contains three layers of neurons (two hidden layers and one output layer). The ﬁrst layer has 10 neurons and tansig function, second layer has ﬁve neurons and tansig function and the output layer contains one neuron with linear function.
2.3. Data pre-processing and post-processing The training data before given to neural network is pre-processed. The pre-processing scheme is as follows. An upper limit of price is set up and following condition is applied.
P U L þ U L Log
ðP P U L Þ
ðP > U L Þ
The upper limit is set to be 70 $/MWh as most of the points are below this limit. To recover the price after limiting the spikes a Post-processing scheme is applied which is as follows
D. Singhal, K.S. Swarup / Electrical Power and Energy Systems 33 (2011) 550–555
if P¼U L UL
U L 10
ðP P U L Þ ðP > U L Þ
MAE ¼ i¼1
jPfi Pai j n
The root mean square (RMS) error is also a standard measure 3. Numerical results The Neural Network is trained with the data of 6 months. It is tested for various days of a particular month. The days include The day with normal trend The day with small spike The day with large spike The resulting price forecasts are described by two different measures, the MAE and RMS error. The mean absolute error (MAE) is a standard measure of accuracy used in forecasting.
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n X 1u RMS ¼ t ðPfi PaiÞ2 n i¼1
where n is the No. of time slots, Pfi is the time slot i predicted value, Pai is the time slot i target value Electricity Price forecasting has been carried out for the three cases corresponding to Price spike with (i) normal trend (ii) small spike and (iii) large spike. Table 2 shows the results with the three case studies. The MAE for electricity price with small spike (0.682) much less than that of the normal trend (2.655) and large spike (9.282), however the RMSE for the price trend with small spike (1.129) is more
Fig. 2. Price forecasting with normal trend.
Fig. 3. Price forecasting with small spike.
Fig. 4. Price forecasting with large spike.
D. Singhal, K.S. Swarup / Electrical Power and Energy Systems 33 (2011) 550–555
than normal trend (0.525) and much less than large spike (4.105). An important inference from the results is that the MAE index alone may provide erroneous results in evaluating the performance of the ANN in forecasting. Both the RMSE and MAE indices should be used together for performance evaluation of the forecasting method used. This suggests the need to identify and use additional performance indices for trend curves with high spike in electricity price. Fig. 2 a shows the forecasted and actual value for normal trend in electricity price. It can be observed that predicted value is very close to the actual value. The neural network is able to track the spike in electricity price and predict the price spike to a good accuracy. Neural Network Forecasting for a short and small spike in electricity prices is shown in Fig. 3. It can be observed that the predicted value is able to track the actual value for small variations at 13th h and 42nd h, but is unable to accurately forecast for the spike at 32nd h. The neural network is able to generalize the forecasting task at major intervals but degrades for local time period. This may be attributed to the deﬁciency in input patterns containing more information about spikes. Feature selection and extraction of spike data during input processing and presentation as a set of input patterns during training may further reduce the error in forecasting to a substantial level. Fig. 4 shows the forecasted and actual value with large spike in electricity prices. Similar to small spike, the neural network is unable to track the large and sudden spike in electricity price. The neural network is able to perform well at all time period, except at the peak duration. Forecasting error during spikes is a major concern for players to broadcast their ‘sell’ and ‘buy’ bids for the sale and purchase of bulk amounts of power during spot pricing of electricity. Further reduction in the forecasting error during price spikes would help the power trading market and independent players with better bidding strategies for efﬁcient operation and increase in savings and social beneﬁt. 4. Conclusions Forecasting electricity price using neural networks in open power markets is presented. Accurate price forecasting is very important for electric utilities in a competitive environment created by the electric industry deregulation. The strong interdependence between load demand and electricity price is considered. Historical information of load and electricity price in forecasting the day-ahead price is presented. A simple price forecasting tool
using multi-layer neural network employing back propagation algorithm has been developed as an aid to the power trading simulator. The neural network model was employed to forecast the market-clearing prices (MCP) of the daily energy market. The forecasting results show that the model is efﬁcient for days with normal trend, however shows a gradual degradation on performance for days with price spikes. The results of the simulation have been tabulated with a less than 16% error on a weekday and a less than 20% error on a weekend. Price forecasting results show that electricity price in the deregulated markets can be forecasted with reasonable accuracy. Electricity price forecasting can be made more accurate by combining several techniques such as fuzzy logic, neural networks and dynamic clustering together. The price for the days with price spikes can be forecasted better by considering the inputs which can explain the reason for spikes so this can be taken into account. References  Song H, Liu CC, Lawarree J, Dahlgren RW. Optimal electricity supply bidding by Markov decision process. IEEE Trans Power Syst 2000;15(2):618–24.  Arroyo JM, Conejo AJ. Multi-period auction for a pool-based electricity market. IEEE Trans Power Syst 2002;17(4):1225–31.  Bunn DW. Forecasting loads and prices in competitive power markets. Proc IEEE 2000;88(2):163–9.  Angelus Alexander. Electricity price forecasting in deregulated power markets. Electr J 2001;14(3):32–41.  Rodriguez CP, Anders GJ. Energy price forecasting in the Ontario competitive power system market. IEEE Trans Power Syst 2004;19(1):366–74.  Park DC, El-Sharkawi MA, Marks II RJ, Atlas LE, Damborg MJ. Electric load forecasting using an artiﬁcial neural network. IEEE Trans Power Syst 1991;6(2):442–9.  Hong Y-Y, Hsiao C-Y. Locational marginal price forecasting in deregulated electricity markets using artiﬁcial intelligence. IEE Proc Generat Trans Distribut 2002;149(5):621–6.  Sapeluk A, Ozveren CS, Birch AP. Pool price forecasting: a neural network application. UPEC 94 Conference Paper 1994;2:840–3.  Szkuta BR, Sanabria LA, Dillon TS. Electricity price short-term forecasting using artiﬁcial neural networks. IEEE Trans Power Syst 1999;14(3):851–7.  Hongming Yang, Xianzhong Duan. Chaotic characteristics of electricity price and its forecasting model. IEEE Can Conf Electr Comput Eng 2003;1:659–62.  Zhengjun Liu, Hongming Yang, Mingyong Lai. Electricity price forecasting model based on chaos theory. International power engineering conference (PEC); 2005. p. 1–5.  García-Martos C, Rodríguez J, Sánchez MJ. Mixed models for short-run forecasting of electricity prices: application for the Spanish market. IEEE Trans Power Syst 2007;22(2):544–52.  Amjady N, Daraeepour A, Keynia F. Day-ahead electricity price forecasting by modiﬁed relief algorithm and hybrid neural network. IET Generat Trans Distribut 2010;4(3):432–44.  Zhi Zhou, Chan WKV. Reducing electricity price forecasting error using seasonality and higher order crossing information. IEEE Trans Power Syst. 2009;24(3):1126–35.