Short-Term Electricity Price Forecasting Using the Empirical Mode Decomposed Hilbert-LSTM and Wavelet-LSTM Models

Abstract

The electricity sector deregulation has led to the formation of short-term power markets where the consumers can purchase electricity by bidding at the electricity market. The electricity market price is volatile and changes are due to change in electricity demand and the bid price at different span of time during the day. The availability of the electricity price forecast is essential for the electricity market participants to make informed decisions. In this paper, the modified LSTM approach, wavelet-LSTM, and Hilbert-LSTM are proposed to predict the electricity price for bidding in the short-term electricity market. The objective is to improve the precision and adaptability of electricity price predictions by utilizing the temporal dependence identification capability of LSTM and the multiresolution analysis capability of the transforms. The proposed models combine these two effective methods in order to capture both the long-term trends and short-term variations present in electricity price time series data. In this approach, the 8-year dataset is used for training the models, and based on this the day-ahead price is calculated and compared with the testing data. The proposed techniques show better performance in terms of rank correlation, mean square error, and root mean square error compared to the existing algorithms of LSTM and CNN-LSTM. The prediction results achieved with wavelet-LSTM and Hilbert-LSTM (1-month dataset of 8 years) are rank correlation 0.9746 and 0.9749, MSE 0.2962 and 0.1363, and RMSE 0.5443 and 0.3692, respectively. The results achieved with the proposed methods are better than the existing forecasting models, and the RMSE for Hilbert-LSTM and wavelet-LSTM techniques is improved by 61% and 43%, respectively, compared to the LSTM method. Also, results are calculated for the complete 8 years all 12 months with Hilbert-LSTM, and the results achieved are rank correlation 0.9645, MSE 0.3876, and RMSE 0.6225. The results achieved with the proposed models are improved in terms of performance parameters compared to the conventional approaches. The proposed models can be used in the day-ahead electricity price forecasting to bid for electricity accurately in the day-ahead electricity market.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The deregulation and liberalization of the energy sector has transformed the electricity sector from the vertically integrated to the decentralized form. The electricity suppliers can competitively sell the electricity in the short-term electricity market, and the consumers are now able to purchase electricity at preferred rates by bidding in the electricity market. The inclusion of the renewable energy resources has made it possible to fulfill the renewable energy purchase obligations that benefit the consumers, suppliers, and the utilities. The reliable operation of the electricity grid requires the utilities to be able to maintain the power supply and demand equilibrium to the scale that is achievable in the day-ahead electricity markets [1, 2]. The different distributed energy resources (DER) and energy storage applications will improve the electricity grid reliability by reducing the peak electricity demand. Also, the demand response (DR) programs are estimated to potentially reduce the increase in the peak demand by 17% by 2050. In addition with the time-dependent electricity pricing schemes, the peak demand will be reduced by 15% [3].

With the advent of technology, there has been a lot of development in the electricity market. The sale of electricity in the electricity market which is mainly long term and medium term is now increasing in the short term as well [4]. This is the result of the inclusion of renewable energy resources in the electricity market and other incentive mechanisms for consumers to participate as part of DR programs. As a result, the electricity market has introduced dynamic pricing schemes and market bidding mechanism [5], where generators can bid their power at competitive prices, and consumers can buy power at reasonable prices. This requires developing a model that can accurately predict the price of electricity for appropriate bidding strategies [6]. It is difficult to store electricity in large quantities. Therefore, whenever it is generated at a generating station, it has to be utilized instantly at the distribution station. As a result, the short-term electricity price is subject to many constraints, and it varies throughout the day. The electricity market in the short term operates by getting the bids from the buyers for the power and the price at which they want to purchase it. Also, by getting the bids from the generators for the power and the price at which they want to sell it. The market system operator then selects the bids based on the price quotes and the power availability, and the supply demand equilibrium determines the market clearing price. This process is entirely unspecified to the participants where they can only decide the price and power bid. Therefore, it is necessary to have a forecast available before bidding in the market.

The competitive bidding from the various participants leads to the market clearing electricity price data to be nonlinear, nonstationary, and volatile. There are various approaches to forecast the electricity price. These are broadly classified as the stochastic methods: techniques that make use of probability function to model the data and forecast the price [7–9]. Then, there are multi-agent models: game theoretic approaches are used to model the consumer pattern [10–12], there are the computational techniques that are the structural models that can be further classified as the techniques that train on data, those which train on pre-defined set of rules, and those that learn and improvise based on certain constraints over the iteration [13, 14]. Then, there are statistical models that separate the data into different components and then do the forecasting using statistical techniques. These are the techniques that are discussed in further sections.

The contribution summary is as follows:

• Firstly, the wavelet Long Short-Term Memory (LSTM) method is presented wherein the wavelet transform (WT) is used to transform the time series electricity price data collected from the electricity market at Indian Energy Exchange (IEX) into frequency-time domain as against the traditional frequency synthesis signal transformation technique as the Fourier transform that transforms the signal into frequency components. In this hybrid architecture, the discrete wavelet transform (DWT) will decompose the electricity price signal into different levels of detail coefficients and approximate coefficient signals that are given as input to the deep learning LSTM network as feature vector set to predict the electricity price.

• Secondly, the Hilbert-LSTM method is presented. Here, the Hilbert transform is used to transform the electricity price signals that are decomposed with the empirical mode decomposition (EMD) technique that will generate different intrinsic mode function (IMF) signals. The IMF signals are Hilbert transformed to generate the analytical signals that are given as a feature vector set to the LSTM network that will forecast the electricity price. The LSTM with their ability to model the long-term dependency in the input data can accurately forecast the time series. Therefore, the innovative aspect of combining the signal decomposition and transformation method with the deep learning method improves the forecasting accuracy than the conventional approach.

• Lastly, the forecasting performance of the wavelet-LSTM and Hilbert-LSTM methods is compared with the traditional LSTM and the convolutional neural network-LSTM (CNN-LSTM) methods for the monthly electricity price dataset, and the results indicate improvement in the prediction accuracy of the models, with Hilbert-LSTM method giving better results in terms of the mean squared error (MSE) and root mean square error (RMSE). Later, the Hilbert-LSTM method is used to forecast the electricity price for the 8-year dataset, and the results are shown to be comparable.

The paper is organized as: Section 2 includes the related work, Section 3 depicts the working methodology, Section 4 showcases the results and discussion, and Section 5 is the concluding section.

2. Related Work

Electricity price forecasting can be done by the data-driven approach, which can be classified into different categories. The first is the time series model, which discusses techniques such as autoregressive (AR), moving average (MA), and auto-regressive moving average (ARMA) [15, 16]. The ARMA model is the combined form of AR and MA models. Its benefit is that it is a comparatively simpler model with fit to the data, whereas the auto regressive integrated moving average (ARIMA) model solves the nonstationary sequence problems [17]. In [18], a hybrid model is used to forecast the electricity prices in the Australian market. The time series models are not only used for prediction but also to process stationary time series, and also these models are predominantly used for short-range forecasts [19].

The second category is the regression models, where regression equation between the independent and dependent variables is used as a prediction model to analyse the market [20]. The regression models are classified depending on the independent variables’ numbers or the correlation among the independent and dependent variable. There are different regression models like linear regression, multilinear regression, etc. [21]. Among these, the hybrid model is used to forecast daily electricity prices in the European electricity market.

The third category is the machine learning models that comprise the three subcategories such as artificial neural network (ANN), support vector machines (SVMs), and decision tree (DT). The ANN is more popular than the other two models, and as far as the energy prediction is concerned, there are many models available, both single models and hybrid models. There are various ANN models, as feed forward neural network (FFNN), backpropagation neural network (BPNN), and recurrent neural network (RNN). Apart from these varieties of deep learning models [22], the SVM is better at handling samples of small sizes and nonlinearity. The vital variant of SVM is least square SVM (LSSVM) that helps in changing the inequality constraints to equality constraints in SVM. Generally, the SVM models exist as hybrid models when it comes to forecasting electricity price or energy price [23]. The DT is another model which is similar to a tree-based hierarchy having models such as the random forests (RFs) and XG-Boost. The advantages of DT are that their computation time is comparatively less, they are insensitive to missing data, and are simple model with strong interpretability [24].

After the collection and gathering of data, the next step is the selection of the forecasting technique. These techniques belong to the domain of artificial intelligence and machine learning. AI is a very versatile field and consists of the study of interdisciplinary fields such as neuroscience, optimization, control theory, economics, and statistics. There are different categories in AI such as machine learning, ANN, and deep learning. Machine learning includes techniques that train from the input data and identify patterns in the data. In machine learning, there are different categories such as supervised learning (SL), unsupervised learning (UL), and reinforcement learning (RL). The SL is a set of methods where their task is to figure out the correlation between the features and the output, given that there is some label associated with the input-output pairs. This part of the dataset that is used for learning is known as the training dataset. The type of output variable decides which problem is the algorithm trying to solve, when the output variable is a numerical type then it is a regression problem, and when it is a categorical variable, it is a classification problem [25, 26]. In SL, there are three basic types of models, namely, linear regression, kernel, and tree-based models. The linear regression models are comparatively simpler models than the kernel models, and offer great interpretability of how the variables interact. The kernel-based models are SVM and Gaussian processes (GP). The author used SVR for forecasting electricity load. The tree-based models are extensively used in load and price forecasting and include methods like regression trees, multiregression trees, classification and regression trees (CART), and gradient boosting DTs [27]. The tree models are interpretable, and handle missing data well, but are prone to overfitting and are normally unstable. Reference [28] used a simple neural network as DNN by embedding seasonality information to forecast the price. Although the time series techniques have been used previously, hybrid wavelet-based ARIMA model to forecast electricity prices has been developed. The price data are decomposed into components by the WT, and the forecasting of these components is done using the ARIMA model [29]. Complicated deep learning architecture like LSTM is used to integrate the seasonal behaviour, a hybrid model based on LSTM and RNN is developed to forecast electricity prices for the next hour [30].

The hybrid CNN-LSTM model is developed by the authors to predict the nonlinear velocity of the river water flow [31]. A similar method is used to forecast solar power, by extracting features of weather conditions using CNN and using LSTM that forecasts the generated power dependent on the weather circumstances [32]. The authors used the ensemble models based on the traditional time series techniques and neural network models to forecast the electricity price [33]. The authors in their paper have used the signal decomposition technique to analyse the trend and seasonality components in the electricity price signal to forecast better [34].

Previous studies have used the time series forecasting and regression models for forecasting the electricity price, but their accuracy is limited due to factors such as the nonlinearity of the data that the regression models cannot model, overfitting of data with price fluctuations, nonstationarity in the data results in inaccurate predictions that can be minimized with time series forecasting methods and by differencing and detrending. Deep learning models are nonlinear models that can model the nonlinear dependencies better than the time series models. Time series models are more structured in estimating the trend, seasonality, and cyclicity components in the data, but their accuracy is limited due to less capability to model the nonlinearity in data. The traditional time series forecasting techniques require the data analysis to select the appropriate order of the model, whereas the LSTM model has inbuilt capabilities to train and generate the model. With the deep learning models, the models are difficult to auto-tune and computationally intensive but are more accurate than the time series models as they are able to intricately train the model from the data. The tuning of the models can be done with the trial-and-error method. The problem of the neural networks can be of the vanishing gradient problem, which can be solved using the LSTM models. The electricity price signal is nonlinear and nonstationary with random variations. Therefore, to predict with better accuracy, the signal decomposition techniques can be used to better model the time-varying frequency content in the time series price signal. Using the signal decomposition techniques, researchers can better understand the long-term and short-term variations and the abrupt fluctuation in the data. The time-dependent frequency analysis of the signals can help to extract the important features in the input data. The hybrid methods of signal decomposition and transformation with deep learning techniques can improve the prediction accuracy. The improved forecasting accuracy can develop better insights for the market participants for bidding appropriately. There are other signal decomposition methods such as variational mode decomposition (VMD), nonlinear signal processing (NPS), and iterative filtering (IF), but the EMD method is used as it is better to decompose the nonlinear and nonstationary electricity price data [35]. Also, even though the VMD method is computationally less intensive than the EMD method, the EMD method does not require a basis function to decompose the signal as it is adaptive to the input signal characteristics. This makes the EMD method widely suitable to different signals. The EMD method is more suitable for low-frequency signals than the VMD method. Although the problem of mode mixing can occur with the EMD method, its different variants can effectively handle the problem.

3. Methodology

By considering the advantages of these approaches, authors propose hybrid approaches like wavelet-LSTM and Hilbert-LSTM to forecast the day-ahead electricity price. Since Hilbert-LSTMs excel in capturing long-range dependencies, they are recognized for their ability to identify broad trends in the time series. In contrast, wavelet-LSTMs are excellent at breaking down time series data into different frequency components, which makes it possible for them to efficiently capture transient fluctuations and abrupt shifts. Wavelet-LSTM can handle nonstationary data of price but the problem of nonlinearity still exists and can be solved by Hilbert-LSTM, which can tackle both nonlinear and nonstationary datasets. In this article, EMD is used to decompose the signal at different IMF levels, and these signals are used for forecasting. To create a complete and adaptable framework for forecasting short-term electricity prices, this novel methodology synthesis seeks to capitalize on the advantages of both methodologies. The proposed approach methodology is explained in the following sections.

3.1. LSTM

A deep learning neural network model known as LSTM is comparable to an RNN [36]. The RNN has a single hidden state that makes it capable of only learning the dependencies in the short term, whereas the complex structure of the LSTM makes it capable to learn short-term and long-term dependencies. Figure 1 shows the structure of an LSTM memory cell, which makes it suitable to forecast the time series data. LSTM has a memory cell made up of the summing and multiplying operators, the tanh and sigmoid functions processing the data taken from different inputs. These together comprise the different gates in the LSTM memory cell. These gates determine the information that is to be removed from, added to, and given as output from the memory cell, after taking the input from the various inputs to the memory cell. The LSTM cell takes three inputs; $C_{t-1}$ is the previous cell state, the previous cell hidden layer output is $h_{t-1}$, and the current input data are $x_t$. The forget gate $f_t$ regulates the information that is removed, the input gate $i_t$ regulates the data that are added, and the output gate $o_t$ regulates the data that are output from the memory cell. LSTM cell uses these gates to control the data to be stored or discarded, which are governed by the following equations:$$\begin{array}{c}\text{(1)}& f_t=\sigma W_f\cdot h_{t-1},x_t+b_f,i_t=\sigma W_i\cdot h_{t-1},x_t+b_i,\\ o_t=\sigma W_o\cdot h_{t-1},x_t+b_o,\\ {\tilde{C}}_t=\tan h{W}_c\cdot h_{t-1},x_t+b_c,\\ C_t=f_t\cdot C_{t-1}+i_t\cdot {\tilde{C}}_t\\ h_t=o_t\cdot \tan h{C}_t,\end{array}$$where σ denotes the sigmoidal function giving output in terms of 0 and 1, $C_t$ is the cell state, and ${\tilde{C}}_t$ denotes the candidate value of the cell state. The LSTM network consists of the neurons having weight matrices for the $f_t$, $i_t$, $o_t$, and ${\tilde{C}}_t$ denoted as $W_f$, $W_i$, $W_o$, and $W_c$, respectively, and $b_f$, $b_i$, $b_o$, and $b_c$ denote the biases, respectively.

[figure(s) omitted; refer to PDF]

3.2. WT

The frequency transform is a method to represent the signal in terms of the frequency content and the amplitude. WT is a technique that gives information about the time and frequency content in the signal. In the WT, a small part of the continuous signal, known as the wavelet, is used to transform the continuous signal. There are different types of wavelets depending upon the symmetry and the function. Consider a signal $xt$ that is integrated under a particular wavelet function. This wavelet is a part of a wave that is used on the signal to get the information of frequency in the time domain.$$\begin{array}{}\text{(2)}& Ta,b=\frac{1}{\sqrt{a}}{\int }_{-\infty }^{+\infty }xt\psi \frac{t-b}{a}dt.\end{array}$$

Equation (2) is the equation for the coefficient for the WT where $a$ and $b$ are real numbers known as scale parameter and translation parameter, respectively. The parameters $a$ and $b$ are varied throughout the process to collect the frequency and amplitude information. The peculiarity of the WT is that it uses time scaling properties to analyse the signal to various scales. This makes the WT suitable to deal with time series data that are nonstationary [37].

In this paper, the DWT technique is used to decompose the electricity price data. For a particular wavelet function as Daubechies, db3 has three vanishing moments and db5 has five vanishing moments. The approximation order corresponds to the number of vanishing moments and will decide the smoothness of the wavelet [38]. For a wavelet function having n vanishing moments, it can approximate polynomials to degree n − 1.

3.3. EMD-Based Hilbert Transform

Hilbert–Huang transform (HHT) is a technique to represent the signal information of frequency in the time domain along with the amplitude or energy of the frequency component as a function of time. In this method, the signal is decomposed into different components using the EMD process. The decomposed component of the signal is known as the IMF. Later, the IMFs are transformed using the Hilbert transform to form a spectrum, that is, the amplitude on the frequency time distribution. The HHT is particularly used to analyse the time series data, as it gives better frequency time resolution for nonlinear and nonstationary data [39].

3.3.1. EMD

Using the EMD method, a time series signal can be decomposed into IMFs, and the IMFs must comply with certain conditions as follows:

• The number of extremes and zero crossings must be equal or should not differ by more than one.

• The average of the envelope characterized by the local maxima and minima should be zero.

• The IMFs should have not less than two extreme values, either minimum or maximum.

The process to compute IMF signal components is shown in Figure 2, and the steps are as follows:

  Step 1. Identify the extremes of the input signal $xt$.

  Step 2. Calculate the bounding envelopes for the extrema points based on the cubic spline interpolation method.

  Step 3. Calculate the mean value based on the upper and the lower envelopes $mt$.

  Step 4. Calculate a new signal $c_{i}t$ = $xt$ − $mt$.

  Step 5. Assign the $i^{\text{th}}$ IMF as the $c_{i}t$ signal.

  Step 6. Calculate the residual signal $rt$ = $xt-c_{i}t$ and set $xt$ as $rt$.

  Step 7. Repeat the previous steps until the last iteration where the IMF will become a signal having zero mean.

  Step 8. The process ends when the monotonous function is attained, that is, $rt$ signal with only one maxima or minima.

[figure(s) omitted; refer to PDF]

After the IMFs are computed, the Hilbert transform is calculated for each IMF that gives the respective amplitudes and frequencies.

The EMD method is a more advanced method of signal transformation. It uses the information of frequency to separate a signal into component signals. The signals are decomposed into IMFs based on the analysed sequence of the input signal. The basis function in EMD is generated from the input data and is therefore suitable for various types of signals. The IMFs generated are more particular and detailed in capturing the frequency content of the signals. Although the EMD is a more sophisticated technique for signal decomposition, it might suffer from the problem of mode mixing during the process, with the IMFs containing different frequencies. This problem occurs if the signal has abnormal number of signal extremes and frequency variations. Still, the results indicate that this problem does not affect the EMD process and the forecasting accuracy is better.

3.3.2. Hilbert Transform

The Hilbert transform of a signal $xt$ is given as [40]$$\begin{array}{}\text{(3)}& x_{h}t={\int }_{-\infty }^{+\infty }\frac{xt}{t-\tau }dt,\end{array}$$here $xt$ and $x_{h}t$ are the complex conjugate pairs, and using the Euler’s identity, $zt$ can be expressed as$$\begin{array}{}\text{(4)}& zt=xt+j{x}_{h}t=at{e}^{j\varphi t},\end{array}$$where $at$ is amplitude and $\varphi t$ is the phase given by$$\begin{array}{c}\text{(5)}& at=\sqrt{{xt}^2{+x_{h}t}^2},\varphi ={\tan }^{-1}\frac{x_{h}t}{xt},\end{array}$$and $\omega t$, the time-dependent frequency is given by$$\begin{array}{}\text{(6)}& \omega t=\frac{d\varphi t}{dt}.\end{array}$$

3.4. Wavelet-LSTM

In Figure 3, the methodology of the proposed technique of wavelet-LSTM is shown. The electricity price data collected from the IEX is processed and normalized. The DWT model transforms the processed data that will give the detailed coefficients and approximate coefficients. These are considered the feature vectors and given as input to the LSTM model. These data are divided into training and testing sets. The LSTM architecture has sequence input layers as the number of features, the learning rate is 0.001, the number of hidden layers is 64, the maximum epoch is 100, and the batch size is 32. Parameters are selected based on the trial error method, and parameter values giving the best results are used for tuning. In wavelet-LSTM decomposition, the Daubechies (db) 3 wavelet family/kernel is used, and the approximate and detailed coefficients' features from the decomposition process are fed to the input layer of LSTM. The LSTM model network trains through the backpropagation and predicts the future price. The model performance is evaluated using the metrics as MSE and RMSE.

[figure(s) omitted; refer to PDF]

3.5. Hilbert-LSTM

In Figure 4, the methodology of the proposed technique of Hilbert-LSTM is shown. The price data collected from the IEX are processed and normalized. The Hilbert transform model transforms the processed data that will decompose the data through the EMD into IMFs. These IMFs are then Hilbert transformed to generate the amplitude and frequency information in the time domain and are considered as the feature vectors that are to be given as input to the LSTM model. These data are divided into training and testing sets. The LSTM architecture has a sequence of input layers as the number of features, the learning rate is 0.001, the number of hidden layers is 64, the maximum epoch is 100, and the batch size is 32. Parameters are selected based on the trial error method, and parameter values giving the best results are used for tuning. The LSTM model network trains through the backpropagation and predicts the future price. The model performance is evaluated using metrics as MSE and RMSE.

[figure(s) omitted; refer to PDF]

4. Results and Discussion

The electricity price data are collected from the IEX [41]. The descriptive statistics for the data are given in Table 1, which can provide statistical insights about the dataset [42, 43], with the mean as 4.86, standard error as 0.0123, median as 3.65, mode as 12, standard deviation and sample variance as 3.24 and 10.52, respectively. The skewness is 1.97, the range is 19.4, and the minimum and maximum values are 0.59 and 20, respectively.

Table 1

Descriptive statistics of the electricity price data.

Descriptive statistic	Value
Mean	4.86 Rs./kWh
Standard error	0.0123 Rs./kWh
Median	3.65 Rs./kWh
Mode	12 Rs./kWh
Standard deviation	3.24 Rs./kWh
Sample variance	10.52 ${\text{Rs}./\text{kWh}}^2$
Skewness	1.97
Range	19.4 Rs./kWh
Minimum	0.59 Rs./kWh
Maximum	20 Rs./kWh

The electricity price signal is decomposed to analyse the trend, seasonality, and residual components. Figure 5 depicts the components in the actual electricity price signal for the specific duration from January 2020 to December 2021 to enhance the readability. It indicates that there is no linear trend annually. The seasonality has been modelled for the weekly time period. The electricity price time series is nonlinear and exhibits random fluctuations. The electricity rate changes at every 15-min time span. The monthly data for all 12 months of the 8 years from January 2015 to December 2022 are collected. Different forecasting techniques such as LSTM, CNN-LSTM, wavelet-LSTM, and EMD-based Hilbert-LSTM are applied to forecast the time series. The models are trained on 80% of the data and tested on the remaining 20% data. Firstly, the 1-month dataset for 8 years is used to evaluate the model performance with the forecasting algorithms.

[figure(s) omitted; refer to PDF]

The electricity price data are available in Rupees per kWh in every 15-min time slot. Then, initially, the dataset is normalized using Z-score normalization. This normalization technique is used here because data are not normally distributed and has a higher variance. The range after standardization can extend to a wider range, as shown in Figure 6 [44]. After the data are normalized first, we apply the LSTM method for forecasting. For this, the data are arranged in a single-column vector. Then, 80% data are used for training purpose, and the remaining 20% are used for testing purpose. The performance of the models is evaluated using statistical parameters as the rank correlation, MSE, and RMSE and calculated using the following equations:$$\begin{array}{c}\text{(7)}& \text{Spearman}’\mathrm{s}\text{\hspace{0.17em}rank\hspace{0.17em}correlation\hspace{0.17em}coefficient}=1-\frac{6{{\displaystyle \sum }}_i{d}_i^2}{n{n}^2-1},\text{MSE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x_i-y_i}^2,\\ \text{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x_i-y_i}^2},\end{array}$$where $d_i$ is the difference between the ranks of $x_i$ and $y_i$ that are the actual and predicted values, and $n$ is the number of samples.

[figure(s) omitted; refer to PDF]

The rank correlation indicates how well the model is trained; the Spearman’s rank correlation coefficient is used to evaluate the rank correlation as it is suitable for the data where the variation in the variables is monotonic and the coefficient is least affected to outliers in the data [45]. The value of the rank correlation is between −1 and +1, with negative values showing inverse correlation and positive values showing direct correlation. In time series forecasting, it nearly indicates the accuracy of the model to forecast the data. Values closer to +1 indicate that the model very accurately fits the actual and the target values. RMSE is a more accurate measure of the accuracy, which is more sensitive to the outlier sample values than MAE, and it is more versatile than the MAPE as it gives the output in similar units, and would not be erroneous for price values of zero. Figure 7 shows the rank correlation for the LSTM model, that is, 0.95223.

[figure(s) omitted; refer to PDF]

Next, the WT is applied, which will give the decomposed coefficients. Here, the db3 wavelet kernel is used to decompose the signal. The approximate signal and detailed signals are shown in Figure 8. Figure 9 shows the rank correlation for data, that is, 0.9746. Figure 10 shows the error evaluation for the wavelet-LSTM model, where MSE and RMSE are 0.29621 and 0.54425, respectively.

[figure(s) omitted; refer to PDF]

The WT basically uses the high-pass filter and low-pass filter to generate the approximate and detailed coefficients. The wavelet function coefficients are used to form these filters that convolute on the input signal to generate these coefficient signals. The filters to three levels are used in the experiment that allows the approximate and detailed coefficients to level 3. The low-pass filter generates the approximate coefficients. The output of the low-pass filter is used as the signal to generate the detailed coefficients, using the high-pass filter at every level. The approximate coefficients represent the overall trend in the input signal, and the detailed coefficients represent the high frequency variations as the noise. This signal decomposition method of WT allows for a better, detailed analysis of the input signal. These approximate and detailed coefficients are given as features to the LSTM network to train and forecast the electricity price. The LSTM network is able to intricately model the pattern of the variation of the component signals.

Next, the EMD-based Hilbert transform technique is used to forecast the electricity price, which takes care of nonlinear and nonstationary signals, as the electricity prices are real-time changing at every time slot. In this, initially, the electricity price signal is decomposed using the EMD process, and the IMF values are calculated. This process will continue till a monotonic function is achieved based on the EMD algorithm. The IMF1 has the highest frequency component of the input signal and the frequency content goes on decreasing with the subsequent IMFs. These IMFs generate a detailed representation of the input signal to multiple levels. The IMFs are Hilbert transformed to form the analytical signals that comprise the real and the imaginary parts. Based on these analytical signals, the amplitude and the phase of the signals are generated. These are given as the features input to the LSTM network to train and forecast the electricity price. In Figure 11, the signal decomposition using the EMD process is presented. Figure 12 shows the rank correlation for data that is 0.9749. Figure 13 shows the error evaluation for Hilbert-LSTM, where the MSE and RMSE are 0.13627 and 0.36915, respectively.

[figure(s) omitted; refer to PDF]

Later, the CNN-LSTM model is used to predict the electricity price that combines the CNN and the LSTM network. The architecture includes the convolution layer to extract features of input data, and the LSTM layer detects time-dependent representations of the derived features. Because the model can figure out both short-term and long-term serial information, it is suitable for univariate time series forecasting. The CNN-LSTM model reads the spatial and temporal pattern of the input data and can forecast the future sequence based on past observations [46]. In the CNN architecture, the convolutional and pooling layers form the basic components. The convolutional layer extracts the features from the input data. During this process, the dimensionality of the convolutional layer output increases. To reduce the dimensionality of the extracted features, the pooling layer is used, which reduces the computational costs during the training process [47].

The results of the proposed methods of Hilbert-LSTM and wavelet-LSTM are compared with the existing LSTM model and the recently developed CNN-LSTM model, for the performance metrics. The simulation results are given in Table 2. The comparison of the performance metrics is given in Figure 14. The MSE and RMSE are the least for the Hilbert-LSTM model and are lower for the wavelet-LSTM model than the CNN-LSTM and LSTM models. The rank correlation is a measure of the accuracy of prediction in terms of the correlation between the predicted and actual values. The rank correlation is highest for the Hilbert-LSTM model and higher for wavelet-LSTM than the CNN-LSTM and LSTM models. The results indicate that the proposed models of Hilbert-LSTM and wavelet-LSTM perform better than the existing CNN-LSTM and LSTM models.

Table 2

Results for the proposed techniques.

Technique	Rank correlation	MSE	RMSE
LSTM	0.9522	0.9140	0.9561
CNN-LSTM	0.9471	0.3140	0.5602
Wavelet-LSTM	0.9746	0.2962	0.5443
Hilbert-LSTM	0.9749	0.1363	0.3692

[figure(s) omitted; refer to PDF]

The statistical validation of the prediction results is performed using the Diebold-Mariano test [48, 49]. The errors in the observed and the forecasted values are measured for the forecasting models. Table 3 shows the results of the statistical test in terms of the $p$ value. The null hypothesis is that there is no significant difference in the predicted results of the forecasting models, and the $p$ values of less than 0.05 indicate that the model in the row predicts higher accuracy compared to the model in the column, which forms the alternative hypothesis.

Table 3

Diebold Mariano test results in terms of the $p$ value.

Model	LSTM	CNN-LSTM	Wavelet-LSTM	Hilbert-LSTM
LSTM	—	0.77712	0.99791	0.99791
CNN-LSTM	0.22288	—	0.99790	0.99789
Wavelet-LSTM	< 0.00209	< 0.00210	—	0.81535
Hilbert-LSTM	< 0.00209	< 0.00211	0.18465	—

Later the Hilbert-LSTM technique is applied to the overall data of 8 years. Here, 90% dataset is used for training, and the remaining 10% dataset is used for testing purpose. Figure 15 shows the rank correlation for data that is 0.9645. Figure 16 shows the error evaluation for Hilbert-LSTM for overall data, where the MSE and RMSE are 0.3876 and 0.62258, respectively.

[figure(s) omitted; refer to PDF]

The result achieved with the proposed approach is presented in Table 4.

Table 4

Results for the proposed technique for 8-year dataset.

Technique	Rank correlation	MSE	RMSE
Hilbert-LSTM	0.9645	0.3876	0.6225

The result shows the effectiveness of each method. The results achieved with Hilbert-LSTM are superior to those of wavelet-LSTM, CNN-LSTM, and LSTM in terms of MSE and RMSE. EMD is used for forecasting because it is a data-driven technique that can effectively capture nonlinear complex patterns of time series data. Electricity prices having complex behaviour depends on seasonality, trends, and irregular fluctuations, which traditional linear models may struggle to capture. EMD can decompose a time series into IMFs, which are components representing different scales or oscillatory modes present in the data. By analysing these IMFs, one can better understand and model the underlying dynamics of the time series.

5. Conclusions

The competitive bidding in the electricity market makes the market clearing electricity price nonlinear and nonstationary. It is necessary to accurately forecast the electricity price for appropriate bidding in the electricity market. To forecast the electricity price, it is essential to use forecasting techniques that can properly model the data and predict the future price. In this paper, the modified deep learning technique is used, which allows modelling of the input data more intricately using the LSTM model. The contribution of this work is that new modified time series forecasting techniques are proposed that improve the forecasting accuracy compared to the existing techniques. The two techniques that are developed are wavelet-LSTM and Hilbert-LSTM. The main task of the WT and the EMD-based Hilbert transform is to transform the electricity price signal by decomposing it to multiple levels in a detailed manner and then giving that as input features to the LSTM network. With the help of wavelet and Hilbert transform, the data are converted to a time-frequency domain with the information of the frequency variation with time. Using other frequency transformation techniques would only provide the frequency information. The advantage of transforming the signal to the frequency domain with time information is that the signal is separated into different components that are then given as input to the LSTM model, which is used to forecast the price values, which in turn allows the LSTM network to model the network more systematically. This method forecasts the electricity price more accurately than it does with the traditional LSTM technique.

The results achieved with proposed approaches, wavelet-LSTM and Hilbert-LSTM (1-month dataset of 8 years), are rank correlation 0.9746 and 0.9749, MSE 0.2962 and 0.1363, RMSE 0.5443 and 0.3692, respectively. Also, results were calculated for the complete 8 years all 12 months with Hilbert-LSTM where 90% dataset is used for training and 10% for testing the results achieved are rank correlation 0.9645, MSE 0.3876, and RMSE 0.6225. The results indicate that the proposed model of EMD-based Hilbert-LSTM can predict the time series with better accuracy than the other existing techniques.

The limitations of this research are that the time series forecasting is univariate and does not include other external influences that can affect the electricity price as the power demand and outdoor ambient temperature. The dataset used in the research is collected from the IEX electricity market and can be similarly used for other electricity markets. The future scope would be to set the parameters of the models using the optimization algorithms for improved training and accuracy. Furthermore, advanced variants of the LSTM can be used to compare the performance and other hybrid decomposition methods can be included to improve the prediction accuracy.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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