Day-ahead electricity price forecasting using back propagation neural networks and weighted least square technique

Abstract

This paper proposes the day-ahead electricity price forecasting using the artificial neural networks (ANN) and weighted least square (WLS) technique in the restructured electricity markets. Price forecasting is very important for online trading, e-commerce and power system operation. Forecasting the hourly locational marginal prices (LMP) in the electricity markets is a very important basis for the decision making in order to maximize the profits/benefits. The novel approach proposed in this paper for forecasting the electricity prices uses WLS technique and compares the results with the results obtained by using ANNs. To perform this price forecasting, the market knowledge is utilized to optimize the selection of input data for the electricity price forecasting tool. In this paper, price forecasting for Pennsylvania-New Jersey-Maryland (PJM) interconnection is demonstrated using the ANNs and the proposed WLS technique. The data used for this price forecasting is obtained from the PJM website. The forecasting results obtained by both methods are compared, which shows the effectiveness of the proposed forecasting approach. From the simulation results, it can be observed that the accuracy of prediction has increased in both seasons using the proposed WLS technique. Another important advantage of the proposed WLS technique is that it is not an iterative method.

Graphical abstract

Keywords

day-ahead electricity markets / price forecasting / load forecasting / artificial neural networks / load serving entities

1 Introduction

Throughout the world, the electricity industry is undergoing lots of changes. Electricity industry is evolving into a competitive and distributed industry in which market forces drive the price of electricity and reduce net cost through increased competition [1]. Proper price forecast can help the participating companies in electricity markets to establish cost effective risk management plans [1]. Presently, market participants have to use various mechanisms to minimize and control the risks as a result of market clearing price (MCP) volatility. If MCP is forecasted properly, load serving entities (LSEs) and generation companies (GENCOs) as main market participating entities can maximize their outcomes and reduce risks further.

Electricity price can be considered as an effective candidate input for short-term load forecasting (STLF) in electricity market environments with the advent of restructuring and deregulation of electrical power systems. In the deregulated power system, customers are charged different rates at different hours of the operating day. To achieve maximum benefit and reduce the expense of using electric energy, some customers adjust their consumption behavior according to the price information. It is expected that some customers would shift their load demand to less expensive hours of the day in order to avoid higher costs. This kind of shift will obviously change the nature of the load demand pattern and form a daily load profile different from the one with a fixed rate. This kind of behavior will introduce the price-based variations into load demand profile.

Electric power industry in many countries throughout the world is changing from the vertically integrated market structure to a decentralized market to create a competitive market. This process of conversion from a cost minimization scheme to a profit maximization scheme is known as the deregulation of electric market. The aim of an electricity market is to decrease the cost of electricity through open competition. Forecasting is used to set the bidding strategies of producers and consumers in the electricity market. In bilateral contracts, the producer and consumer usually agree to do energy transaction a few months before the actual dispatch. So, by means of a reliable daily electricity price forecast, the producers are able to set good bilateral contracts. Therefore, accurate price forecasting is very important for producers to maximize their profits, for consumers to maximize their utilities and for both producers and consumers to negotiate in a bilateral contract. This hourly price forecasting is known as short-term price (ST) forecasting. The ST forecasting has more error in comparison to STLF. The price signal presents the following characteristics: high frequency, non-constant mean and variance, calendar effect, multiple seasonality, high volatility, high percentage of unusual prices, due to which the accuracy of forecasting model reduces to a greater extent in respect to load forecasting accuracy.

The price forecasting scheme can be broadly divided into the time series analysis and simulation approach. The simulation approach uses the power system equipment and their cost information such as heat rates or other fuel efficiencies. Besides, to cover various possible situations, such as equipment outages, Monte-Carlo type stochastic simulations are often processed. Because of the requirement of large amount of data on existing equipment, the simulation method is very complex but it can give a good result if used by market regulators and operators who have an authority of collecting precise equipment and operational information. The time series approach can be classified into linear regression-based models and nonlinear heuristic models. Regression-based models are auto regressive moving average (ARMA) models, and its extension, auto regressive integrated moving average (ARIMA) models, and their variants. These models are used for modeling and forecasting the changing price itself, while the generalized auto regressive conditional heteroskedasticity (GARCH) is used to model the volatility of prices. Nonlinear heuristic models are based on artificial neural networks (ANN), which are known to represent any input-output data relations with a proper structure of internal connections.

Reference [2] proposed two new price forecasting mechanisms based on the time series analysis with the dynamic regression model and transfer function model. Reference [3] focused on day-ahead (DA) price forecast of a daily electricity market using ARIMA models. An approach to forecast the DA electricity prices based on wavelet transform and ARIMA models was proposed in Ref. [4]. A hybrid approach that combines ARIMA and ANN models for predicting ST electricity prices was presented in Ref. [5]. A method based on fuzzy neural network used for ST price forecasting of electricity markets was suggested in Ref. [6]. This fuzzy neural network (NN) has an inter-layer and feed-forward architecture with a new hypercube training mechanism. A hybrid model used for ST electricity price forecasting which is the combination of wavelet transform, particle swarm optimization (PSO), and adaptive-network-based fuzzy inference system was developed in Ref. [7]. Here, wavelet is used to make the ill-behaved price series to a set of better-behaved constitutive series, and PSO to optimize the parameter of membership function.

The ANN based approach was proposed in Ref. [1] to forecast the 24 locational marginal prices (LMPs) of a DA electricity market. To select the more appropriate input features, considering the lagged prices and loads, the sensitivity analysis is used in this paper. Reference [8] proposed a composite model for adaptive ST electricity price forecasting using the ANN in deregulated power market. The factors impacting price forecasting, including time, load, reserve, and historical price factors are put forward. In Ref. [9], a NN based model was proposed to forecast market-clearing prices (MCP) of daily energy market. The forecasting results using this approach show that the model is good for the days with normal trend, but shows a gradual degradation on the performance for the days with price spikes. ST forecasting power of 12 time series models for the electricity prices, in the two markets for various market situations were proposed in Ref. [10]. The forecasting results lead to the conclusion that models with system load demand as the exogenous variable generally perform better than the pure price models. Reference [11] presented the experience of Spanish transmission system operator, in forecasting the electricity load. A NN model to forecast the next-week prices in electricity market of Spain was proposed in Ref. [12]. In Ref. [13], an overview of different electricity price-forecasting techniques was presented and the key issues were analyzed.

In Ref. [14], a discrete cosine transforms (DCT) input featured feed forward neural network (FFNN) model had been proposed for price forecasting. It has been presented that the price forecasting results obtained with DCT-FFNN model are close to the state of the art models and they can be achieved with less computation time. Reference [15] presented a hybrid evolutionary adaptive methodology for ST price forecasting, i.e., between 24 h and 168 h ahead, by combining the mutual information, wavelet transform, PSO, and adaptive neuro-fuzzy inference system. Reference [16] proposed an improved forecasting model that detaches high volatility and daily seasonality for electricity price of New South Wales in Australia based on empirical mode decomposition, seasonal adjustment and autoregressive integrated moving average. A new short-term electricity price forecasting scheme based on a state space model of the power market was developed in Ref. [17]. Here, the Gauss-Markov process is used to represent the stochastic dynamics of the electricity market. In Ref. [18], a new price forecast method was proposed, which is composed of a modified version of relief algorithm for feature selection and a hybrid neural network for prediction. The objective of Ref. [19] was to accurately forecast the short-term loads using discrete wavelet transform in combination with ANN/ support vector machine.

This paper presents a conventional learning algorithm in the area of electricity price forecasting. It then applies a new linear weighted least square (WLS) technique to this electricity price forecasting problem to reduce the error. It focuses on day-ahead (DA) short-term decisions associated with the centralized/pool-based electricity markets. Hence, it takes into consideration the forecasting of DA electricity prices. For each day of the week, 24 h price forecasts must be calculated. Since DA electricity price forecasting is very important for generation companies (GENCOs) and load serving entities (LSEs), it proposes a highly accurate and efficient forecasting tool which is tested on the Pennsylvania-New Jersey-Maryland (PJM) interconnection market data. The state estimation (SE) based model and AI models are examined comprehensively, and the detailed classification is presented.

2 Electricity price forecasting

With the introduction of power system deregulation into electric power industry, the price of electricity has become the focus of all activities in the power market. Based on its specific application, price forecasting can be categorized into short-term, mid-term and long-term forecasting. In recent years, electricity has also been traded as a commodity in various markets.

2.1 Implementation of electricity price forecasting using back propagation neural networks (BPNN)

Recently, the application of ANNs to electricity price forecasting has attracted a lot of attention. The availability of historical load demand and price data on utility databases makes this area highly suitable for ANN implementation. ANNs are able to learn the relationship among past, current, and future prices and load demands combining both the regression and time series methods. As it is the case with time series approach, ANN traces previous price patterns and predicts a price pattern using the recent price data. ANN is able to perform nonlinear modeling and adaptation. It does not require the assumption of any functional relationship between the price and weather variables in advance.

In this paper, the PJM interconnection system load demand and price data are considered for electricity price forecasting. The data of four months are collected and the network for the winter season and spring season is trained, separately. The forecasted results for the winter and spring seasons are presented. The typical BPNN structure for the electricity price forecasting is a 3-layered network, with the nonlinear sigmoid function as the transfer function. The construction of NN involves network properties, node properties, and system dynamics.

The three layers include the input, the hidden and the output layer. The connection weights can be real numbers or integers. These weights are adjustable during network training, but some can be fixed deliberately. Once the training is finished, all of them should be fixed.

2.1.1 Node properties

The activation function used is the sigmoidal activation function at the nodes of hidden and the output layers. At the nodes of the input layer, a linear activation function is used, and is expressed as

(1) f(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+e^{-x}}}.

The activation levels at the nodes can be continuous across the range [0, 1].

2.1.2 System dynamics

The weight initialization method is specific to the selected NN model. The initial weights are generated randomly in the range of [(–0.5 to+ 0.5) or (‒1 to+ 1)].

2.1.3 Scaling

The inputs and outputs are normalized with respect to their lower and upper limits in each training pattern.

(2) X_{\mathrm{norm}}={\displaystyle \frac{X_i-X_{\min }}{X_{\max }-X_{\min }}}\times \mathrm{0.8}+\mathrm{0.1.}

In this paper, the back propagation algorithm (BPA) is used for training the NN. The implementation of BPA involves a forward pass through the network to estimate the error, and then a backward pass will modify the synapses (i.e., weights) to decrease the error. The brief description of BPNN is presented in Appendix.

2.1.4 Simulation

Using the trained NN, the forecasting output is simulated using the test input patterns.

2.1.5 Post processing

The NN output needs de-scaling i.e., de-normalization to generate the desired forecasted load demands.

(3) X_{\mathrm{denorm}}={\displaystyle \frac{X_{\mathrm{norm}}-\mathrm{0.1}}{\mathrm{0.8}}}\times (X_{\max }-X_{\min })+X_{\min }.

2.2 Implementation of electricity price forecasting using WLS technique

The SE processes the raw measurements using the WLS technique and produces the reliable data. The SE technique requires more measurements than the number of unknowns. The objective of the SE is to provide a view of real time power system conditions and to provide a consistent representation for power system security analysis. To solve the SE problem, the least square curve fitting technique is used, where the sum of the squares of the difference between measured and calculated values (i.e., errors) is minimized. The mathematical modeling for the linear measurement equations is given by [20]

(4) [Z\mathrm{]=[}A\mathrm{][}X\mathrm{]+[}\phi ].

Equation (4) can be modified as [21]

(5) A_{\mathrm{eff}}X=Z_{\mathrm{eff}},

where A _eff = A ^T R ⁻¹ A of dimension (n ₁× n ₁), n ₁ is the number of state variables, Z _eff = A ^T R ⁻¹ Z and R ⁻¹ is the weight (diagonal) matrix of size (m ₁× m ₁), m ₁ is the number of measurements.

By calculating A _eff and Z _eff, the problem of SE would become a simple problem of solution of a set of simultaneous equation (i.e., Eq. (5)), and

(6) \widehat{X}=A_{\mathrm{eff}}^{-\mathrm{1}}{Z}_{\mathrm{eff}},

where \widehat{X} is the estimate. Hence, the WLS SE technique is a digital algorithm used to process all raw measurements using the WLS technique and to evaluate A _eff and Z _eff and then \widehat{X}. In this paper, the WLS technique is used to forecast the day-ahead electricity price [22]. The algorithm for electricity price forecasting using the WLS technique is as follows:

Step 1: (a) Read the number of training patterns (m) and the number of rows in the weight matrix from the input layer to the output layer (n).

(b) Read A(i, j), where j is from 1 to n, and i is from 1 to m.

(c) Read Z(i), where i is from 1 to m, and Z is the target output.

(d) Read sigma (i), where i is from 1 to m.

(e) Read A _test (i, j), j is from 1 to n, and i is from 1 to the number of testing patterns.

Step 2: Form A ₂ matrix.

(7) A_2=A^T\times R^{-\mathrm{1}}.

Step 3: Determine Z _eff

(8) Z_{\mathrm{eff}}=A_2\times Z.

Step 4: Determine A _eff

(9) A_{\mathrm{eff}}=A_2\times A.

Step 5: Solve x using,

(10) A_{\mathrm{eff}}\times X=Z_{\mathrm{eff}}.

Step 6: Determine Z _test using,

(11) Z_{\mathrm{test}}=A_{\mathrm{test}}\times X,

where Z _test is the target outputs obtained.

2.3 Error analysis

To assess the prediction accuracy of the proposed BPNN and WLS techniques, various statistical measures are utilized. In this paper, the mean average error (MAE) and the mean absolute percentage error (MAPE) [19] are used. The daily MAE and MAPE are expressed by

(12) {\mathrm{MAE}}_{\mathrm{day}}={\displaystyle \frac{\mathrm{1}}{\mathrm{24}}}{\displaystyle \sum }_{h=\mathrm{1}}^{\mathrm{24}}\left|P_h^{\mathrm{actual}}-P_h^{\mathrm{forecasted}}\right|,

(13) {\mathrm{MAPE}}_{\mathrm{day}}={\displaystyle \frac{\mathrm{1}}{\mathrm{24}}}{\displaystyle \sum _{h=\mathrm{1}}^{\mathrm{24}}\frac{\left|P_h^{\mathrm{actual}}-P_h^{\mathrm{forecasted}}\right|}{P_{\mathrm{1}h}^{\mathrm{actual,24h}}}},

where

(14) P_{1h}^{\mathrm{actual,24h}}={\displaystyle \frac{\mathrm{1}}{\mathrm{24}}}{\displaystyle \sum _{h=\mathrm{1}}^{\mathrm{24}}\left|p_h^{\mathrm{actual}}\right|}.

3 Results and discussion

The proposed price forecasting technique has been implemented on the PJM data [23]. The winter and spring data are collected from the PJM website [23].

3.1 Input variables

Different sets of lagged prices and load demands have been presented as input features for price forecasts in the electricity markets. Keeping the daily and weekly periodicity and trend of electricity price signal in mind, the set of input variables are

{P _{h -1}, P _{h -2}, P _{h -3}, P _{h -24}, P _{h -25}, P _{h -47}, P _{h -48}, P _{h -49}, P _{h -71}, P _{h -72}, P _{h -73}, P _{h -95}, P _{h -96}, P _{h -97}, P _{h -119}, P _{h -120}, P _{h -121}, P _{h -143}, P _{h -144}, P _{h -145}, P _{h -167}, P _{h -168}, L _h, L _{h -1}, L _{h -22}, L _{h -25}, L _{h -48}, L _{h -49}, L _{h -72}, L _{h -73}, L _{h -96}, L _{h -97}, L _{h -120}, L _{h -121}, L _{h -144,} L _{h -145}, L _{h -168}}.

Apart from these 37 inputs, two other inputs i.e., the hour of the day and the day of the week are also included in the input set. Here, 39 input nodes, 57 hidden nodes and one output nodes give the best forecast for the actual signal. Various configurations are tested with different input and output NNs. But the node structure with (39-57-1) gives the best simulation results. These 39 inputs are used at the input layer and are considered as input features for this paper. The considered input features are the lagged prices and load demands recommended in several papers. P _h, i.e., price at hour h is the single output of the training inputs. When the price at hour h is to be forecasted, it is used as P _{h -1} for the price forecast of the next hour, and this is repeated till the price of the whole forecast horizon (here, next 24 h) is forecasted. As explained, the number of hidden nodes (N _H) in the BPNN is 57 (i.e., N _H is 57). The learning rate (h) is selected as 0.4, and the momentum rate (a) is selected as 0.5. These are the values recommended for generalized delta rule (GDR) training mechanism. The larger values of momentum and learning rates may accelerate the learning process, but at the same time, the convergence of GDR may lose. The larger values of the momentum rate may result in the oscillation of training error.

Training period

If the functional relationships of a signal slowly vary with respect to the time, a long history of signal can be considered as the training period, which results in a large number of training samples. Moreover, the obtained results from one training phase, such as the weights of an ANN, can be used for many forecasts. However, as explained earlier, the functional relationships of LMP vary fast compared to the hourly load. Therefore, the training period of LMP is more limited than the hourly load demand. Suppose, if the training period is very short, the ANN cannot derive all functional relationships of the LMP due to the small number of training samples. In Refs. [24,25], the last 48 days have been presented as the training period for the ANN based price forecasting. Hence, the ANN is trained with the training samples in the form shown in Table 1. This will forecast the LMP of the next hour, which are unseen for the ANN. When the LMP of an hour is predicted, it is used to predict the LMP of the next hour, and this will be repeated until the LMP of the next 24 h is forecasted. For forecasting the price of the 24 h of today at one hour interval, the price of the previous 48 days is used for training the NN.

The data of two months (winter season: January and February 2014) are collected from the PJM system website in which the data 59 days are made available. For this, at one-hour interval step, a total of 59 × 24= 1416 data elements are possible. Similarly, the data of two months (spring season: April and May 2014) are collected from the PJM website. To train the ANN with the architecture of 39 input neurons, 57 hidden neurons and one neuron at the output layer are used. For each price at an interval h, the data element value at that interval is selected as the single target output, and the data elements from the previous 7 days are considered as the input feature, as listed in Table 2.

Notes: P _h is the target/price output at interval h. D is the day of forecast. L _{h -1} to L _{h -168} are 15 inputs to the input layer as shown in Table 2, selected from the load data elements. P _{h -1} to P _{h -168} are the selected 22 inputs to the input layer as shown in Table 2.

The data of the previous 7 days are used for each price element value P _h for training the network. The data of January 1, 2014 to January 7, 2014 are set aside as buffer and the actual training is initiated from the first hour of January 8, 2014. The data from January 8, 2014 to February 21, 2014 are used for training the selected network for forecasting the load demand for 24 h at the hourly interval of remaining days of February 2014. A total of 168 data elements are considered as the test data elements.

Let l ₁, l ₂, l ₃,…, l ₂₄ are the loads and p ₁, p ₂, p ₃, …, p ₂₄ are the prices for 24 h of each day at hourly interval for each day in January and February in 2014.

If the price p ₁₅ of January 8 is taken as the output/target (P _h), the input elements would be

\mathrm{January\; 8:}{p}_{\mathrm{14},}{p}_{\mathrm{13}},p_{\mathrm{12}},l_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 7}:p_{\mathrm{14}},p_{\mathrm{13}},l_{\mathrm{17}},l_{\mathrm{14}}

\mathrm{January\; 6}:p_{\mathrm{15}},p_{\mathrm{14}},p_{\mathrm{13}},l_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 5}:p_{\mathrm{15}},p_{\mathrm{14}},p_{\mathrm{13}},l_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 4}:p_{\mathrm{15}},p_{\mathrm{14}},p_{\mathrm{13},}{l}_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 3}:p_{\mathrm{15}},p_{\mathrm{14}},p_{\mathrm{13},}{l}_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 2}:p_{\mathrm{15}},p_{\mathrm{14}},p_{\mathrm{13},}{l}_{\mathrm{15}},l_{\mathrm{14}}

\mathrm{January\; 1}:p_{\mathrm{14}},p_{\mathrm{13}},l_{\mathrm{15}}

The electricity price forecasting results are presented for the ANN technique i.e., BPNN and the linear WLS technique. Figure 1 depicts the actual price and the forecasted price using the ANN and WLS technique. From Fig. 1, it is clearly observed that the forecasted price obtained using WLS technique is better than that of the BPNN.

Table 3 presents the forecasting errors for winter weeks using the ANNs and the proposed WLS technique. From Table 3, it can be observed that the MAPE obtained from the WLS technique are lesser compared to the MAPE obtained from the ANN.

Figure 2 demonstrates the actual and forecasted electricity prices using the ANN and the proposed WLS technique. The forecasted price obtained using the WLS technique is very close to the actual electricity price.

Table 4 lists the electricity price forecasting for spring weeks using the ANN and WLS technique. The MAPE obtained with the proposed WLS technique is lesser compared to the MAPE obtained using the BPNN.

Table 5 tabulates the weekly MAPE obtained from the BPNN and the proposed WLS technique. The MAPE_week obtained using the ANN during winter weeks is 11.2631, whereas using linear WLS technique is 10.4529. From Table 5, it is clearly noticed that the MAPE_week obtained using the proposed WLS technique is lesser compared to BPNN.

From the above simulation results, it can be observed that the prediction during the winter weeks is not as accurate as that in spring weeks, due to the fact that the electricity price in winter is highly nonlinear. The locational marginal price (LMP) is dependent on a large set of factors such as previous LMPs, hourly loads and temperatures. In this paper, the temperatures which have effect on the accuracy of the prediction in winter season are not considered. From the results, it is also observed that by using the proposed WLS technique, the accuracy of prediction has relatively increased in both the seasons. Another important advantage of this method is that it is not an iterative method.

4 Conclusions

The LMP forecast is an important and difficult task due to the high volatile characteristic of the LMP signal. This paper proposes a new price forecasting tool using the linear WLS technique. The data from the PJM interconnection for winter and spring seasons have been considered. The DA electricity prices have been forecasted using the BPNN and WLS techniques. From the simulation results, it can be observed that the accuracy of prediction has increased in both the seasons using the proposed WLS technique. Another important advantage of the proposed WLS technique is that it is not an iterative method. Considering the weekdays, weekends and holidays, and clustering each day into three different clusters of peak hours, normal hours and off-peak hours, the forecasting of the electricity prices for these clusters separately could yield good accuracy, which is the scope for future research.

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