An Integrated Framework for Electricity Price Analysis and Forecasting Based on DROI Framework: Application to Spanish Power Markets

Abstract

Against the backdrop of electricity market liberalization and deregulation, accurate electricity price forecasting is critical for optimizing power dispatch and promoting the low-carbon transition of energy structures. However, electricity prices exhibit inherent complexities such as seasonality, high volatility, and non-stationarity, which undermine the efficacy of traditional forecasting methodologies. To address these challenges, this study proposes a four-stage Decomposition-Reconstruction-Optimization-Integration (DROI) framework, coupled with an econometric breakpoint test, to evaluate forecasting performance across distinct time segments of Spanish electricity price data. The framework employs CEEMDAN for signal decomposition, decomposing complex price sequences into intrinsic mode functions to retain essential features while mitigating noise, followed by frequency-based data reconstruction; integrates the Improved Sparrow Search Algorithm (ISSA) to optimize initial model parameters, minimizing errors induced by subjective factors; and leverages Convolutional Neural Networks (CNN) for frequency-domain feature extraction, enhanced by an attention mechanism to weight channels and prioritize critical attributes, paired with Long Short-Term Memory (LSTMs) for temporal sequence forecasting. Experimental results validate the method’s robustness in both interval forecasting (IPCP = 100% and IPNAW is the smallest, Experiment 1.3) and point forecasting tasks (MAPE = 1.3758%, Experiment 1.1), outperforming naive approaches in processing stationary sequence clusters and demonstrating substantial economic utility to inform sustainable power system management.

1. Introduction

Amid the ongoing liberalization and deregulation of the power sector, accurate electricity price forecasting has become a fundamental requirement for effective energy management and market operations [1]. Electricity prices in competitive markets tend to fluctuate sharply and display complex temporal patterns. They are characterized by high-frequency movements, non-stationary means and variances, pronounced daily and weekly seasonality, calendar effects associated with weekends and holidays, and recurrent price spikes [2]. Furthermore, structural challenges such as transmission bottlenecks, real-time supply–demand imbalances, and the growing share of renewable energy sources can trigger brief but sharp price spikes. These complex dynamics increase the complexity of forecasting models and heighten the risk of prediction errors, which can significantly affect power plant operators, market decision-makers, and participants [3]. Accurate electricity price forecasting is therefore critical for energy suppliers, market operators, and participants. It enables managers to analyze future electricity demand and develop long-term energy plans while supporting market participants in risk management and purchasing decisions. For governments, it provides an essential basis for energy policy formulation; for suppliers, it assists in designing optimal bidding strategies; and for consumers, it helps them plan purchases to maximize value and minimize costs. As such, achieving precise short-term electricity price forecasts and mitigating market risks from price uncertainty remain central to current research efforts.

Mainstream electricity price forecasting methods can be categorized into three types (1) econometric models, (2) machine learning models, and (3) hybrid models. In the early stage of research, both domestic and international research predominantly relied on mathematical or econometric models. Among these, the autoregressive integrated moving average (ARIMA) model has been widely applied in electricity price forecasting. For instance, Contreras et al. developed an ARIMA model to forecast prices in the Spanish and Californian electricity markets. ARIMA combines two core components, autoregression (AR) and moving average (MA), and achieves data smoothing through differencing operations. Although ARIMA provides good interpretability, it often exhibits relatively large mean percentage errors, with daily average errors ranging from 4% to 10%, which are generally higher than those of machine learning models [4]. To address the limitations of ARIMA model under high volatility, Zhou et al. introduced an interval forecasting approach for hourly market clearing price (MCP) prediction in the California electricity market [5]. In contrast, Girish applied the Generalized Autoregressive Conditional Heteroskedastic (GARCH) based model to forecast hourly electricity prices in India. GARCH models assume that the moments of a time series are non-constant, implying that the error term, defined as the difference between observed and predicted values, may not have a zero mean or constant variance, unlike ARIMA models. Instead, the error term is treated as serially correlated and modeled using an AR process. This makes GARCH models well-suited for capturing implied volatility in time series data, particularly during price spikes. GARCH generally outperforms ARIMA in forecasting, except during periods of low volatility [6]. Despite the strong performance of time series techniques like GARCH, their reliance on linear modeling limits their ability to predict highly nonlinear behaviors and sudden price fluctuations [7]. As a result, researchers have increasingly adopted machine learning methods which have advantages in fitting nonlinear features.

Among machine learning techniques, neural network models have been extensively applied to electricity price forecasting. Compared to traditional statistical approaches, neural networks exhibit superior nonlinear fitting capabilities and adaptability, making them particularly advantageous for handling high-volatility price forecasting. In recent years, artificial neural networks (ANNs) have gained attention for their ability to capture complex nonlinear relationships between input and output datasets [8]. For example, Li and Guo utilized a back propagation (BP) neural network combined with system marginal price (SMP) and dynamic clustering weights to forecast historical marginal price data from the PJM Interconnection (PJM) electricity market in the United States [9]. BP networks can approximate any function with finite discontinuities, given a sufficient number of neurons in the hidden layer. Building on this, Tang and Gu employed backpropagation to update weights and biases. However, BP networks are highly sensitive to noise, requiring rigorous data preprocessing [10]. In contrast, Extreme Learning Machines (ELMs), with random initialization and minimal parameter optimization, are somewhat more robust to noise, making them popular in electricity price forecasting. Shrivastava et al. applied the ELM model to data from the Ontario and PJM markets, demonstrating higher generalization capability and significantly faster processing speeds compared to traditional neural network algorithms [11]. Nevertheless, as a feedforward neural network, ELM struggles to capture temporal dependencies.

The advancement of deep learning (DL) in electricity price forecasting has provided researchers with innovative approaches for time series forecasting. Among DL models, LSTM demonstrates excellent performance in handling nonlinear complex problems and time series data [12], while CNN excels in high-dimensional feature extraction [13], making these models widely applicable. Cantillo-Luna et al. employed a stacked LSTM combined with a time2vec layer to forecast electricity prices up to 8 h ahead, outperforming advanced statistical models like SARIMA and Holt-Winters [14]. Similarly, Abedinia et al. utilized CNN-based approaches to forecast electricity prices in the PJM and mainland Spain markets, achieving favorable results [15]. However, CNNs face limitations in capturing global information and long-distance dependencies. To address this, Shejul et al. integrated CNN and LSTM models for short-term electricity price forecasting, yielding improved forecasting accuracy [16]. Additionally, Pourdaryaei et al. applied a CNN algorithm augmented with an attention mechanism to simulate the Ontario electricity market in 2020, achieving superior results [17]. LSTM stands out in capturing patterns and long-term dependencies in nonlinear time series data, while CNN is effective at identifying local features. The squeeze-and-excitation (SE) attention mechanism further enhances CNN by adaptively weighting each channel, enabling it to focus on globally important features. This integrated framework substantially improves forecasting accuracy and overall performance.

In forecasting research, it is widely acknowledged that no single method excels in all scenarios. To enhance forecasting accuracy, integrated models have become a common approach in electricity price forecasting. These models frequently incorporate (1) decomposition modules and (2) optimization modules to improve robustness. Decomposition algorithms, by mitigating the non-stationary characteristics of time series data, are extensively utilized in hybrid methods for electricity price forecasting [18]. A key advantage of the EMD series decomposition method is its inherent capability to handle non-stationary and nonlinear processes [19,20]. Afanasyev et al. applied CEEMDAN in power price forecasting, demonstrating superior performance compared to standard EMD and wavelet decomposition methods [21]. Another critical factor impacting the performance of deep learning models is improper hyperparameter tuning. To address this, optimization algorithms are increasingly integrated into forecasting models [22,23]. These algorithms enable the identification of optimal model parameters, thereby enhancing both forecasting accuracy and model robustness. Literature research in the field of electricity price forecasting is listed in

Table 1. Literature research in the field of electricity price forecasting.

Categories	Methods	Refs.	Advantages	Disadvantages	MAPE(%)
Econometric models	ARIMA	[4,5]	It has strong interpretability	The percentage error is relatively large	2.2573
	GARCH	[6]	Compared with ARIMA, it effectively evaluates the implied volatility within a time series.	1. For periods of low volatility, the forecasting is not as effective as ARIMA	not applicable
				2. The percentage error is relatively large
Machine learning models	BP	[9,10]	Has the ability to approximate any function with finite discontinuities with arbitrary precision	1. More sensitive to noise	2.9536
				2. Highly dependent on the choice of hyperparameters
	ELM	[11]	Higher generalization ability and extremely fast speed	Difficult to capture the dependence of time	2.6173
	LSTM	[14,24]	Hyperparameter adjustment is difficult	Excels at extracting patterns and understanding long-term dependencies from nonlinear time series data	2.0606
	CNN	[17]	Maybe limited in processing global information	Effective at capturing local features	2.9060
Hybrid models	Decomposition or optimization	[19,20,21,22,23,25]	1. Facilitate feature extraction 2. Reduce modal noise 3. Improve forecasting accuracy	Increased model complexity	1.3758

Note: The selected dataset exhibits significant volatility after 2021, and the GARCH model is not applicable to this dataset.

.

Short-term electricity price forecasting is pivotal to stable power market operation, yet it remains a formidable task due to inherent price volatility and intricate influencing factors. Conventional econometric models and standalone machine learning approaches often fall short in two key areas: effectively extracting critical trend information from raw data and optimizing model hyperparameters, which ultimately restricts their forecasting accuracy. To address these gaps, this paper employs a decomposition -reconstruction -optimization -integration (DROI) framework to address the challenge of electricity price fluctuations. CEEMDAN is applied for data decomposition, with the resulting modules reconstructed using the t-test on the original sequence. After consolidating the decomposed frequency data, the CNN-SE Attention-LSTM (CSL) is utilized to weight and amplify trend information, enhancing feature extraction and improving forecasting accuracy. ISSA is incorporated for global optimization of the data model, determining optimal parameters such as the learning rate, L2 regularization coefficient, and the number of neurons in the hidden layer. Additionally, a residual test ensures comprehensive extraction of trend components. Both interval forecasting and deterministic forecasting methods are applied to compare the performance of econometric models with standalone machine learning predictive models. Results from six experiments demonstrate that the CEEMDAN-ISSA-CNN-SE Attention-LSTM (CSCSL) hybrid model delivers superior performance in short-term electricity price forecasting.

The main innovations of this study are summarized as follows:

(1) To mitigate errors stemming from human factors, the ISSA optimization algorithm is introduced to determine key model parameters, including the initial learning rate, regularization coefficient, and the number of neurons in the hidden layer. These parameters are optimized by minimizing the RMSE in the forecasting model. Furthermore, Lévy flight is incorporated to avoid local optima and achieve a global search, significantly enhancing the model’s stability. This approach not only improves the model’s forecasting performance but also provides a robust foundation for future parameter selection.

(2) To enhance the feature extraction capability of the model, an attention mechanism is incorporated, leading to the development of the convolution-time-loop hybrid deep learning model (CSL) based on the attention mechanism for forecasting. In the CSL forecasting model, CNN provides local perception and parameter sharing, effectively capturing local patterns in input sequences and reducing model complexity. SE Attention assigns weights to different parts of the sequence based on contextual information, accounting for correlations within the input data. LSTM, as a core component, captures long-term dependencies in sequences and demonstrates memory capabilities. This method excels in nonlinear fitting and is particularly suited for short-term electricity price forecasting tasks with inherent volatility.

(3) To improve the nonlinear fitting ability of the forecasting model, the decomposition and reconstruction methods are introduced for electricity price data processing. CEEMDAN is used to decompose the data into multiple modes, automatically adjusting parameters based on signal characteristics to effectively eliminate noise. Subsequently, the model is reconstructed according to the data’s frequency states using an independent sample t-test. This approach enhance the model’s ability to process complex features. Using this method, a single time series is decomposed into three frequency states: high, medium, and low frequency. Predicting each frequency state separately enhances the model’s forecasting performance and reduces forecasting error.

(4) To increase the interpretability of the model, the Chow test is employed to partition the dataset in conjunction with economic events. Discontinuity points are identified based on significant socio-economic events, and the Chow test is applied to confirm the presence of structural breaks. By conducting these experiments, the time series data is segmented into multiple datasets, aiding in the evaluation of the model’s adaptability to different scenarios and its robustness. This approach provides valuable insights and serves as a reference for future model selection and improvement.

The remainder of this paper is organized as follows. In Section 2, the basic principles of the proposed model and correction method are introduced, respectively. In Section 3, the preparation of the experiment is introduced, including dataset reconstruction, benchmark model, evaluation index, model parameters, dataset division and experiment arrangement. In Section 4 and Section 5, the forecasting performance and robustness of the model are studied from two perspectives. In Section 6, the experimental results are summarized and future research prospects are discussed.

2. Methodology

This section begins with an introduction to the basic principles of the decomposition method, followed by the optimization algorithm and forecasting model (with relevant formulas provided in the Appendix A and Appendix B).

2.1. The Decomposition Method

For power price data characterized by high volatility, the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) can effectively decompose the original highly fluctuating data into multiple smoother components [26]. This method combines the modal information of each component and employs econometric t-tests to reconstruct the component characteristics. This process aims to minimize data variability and suddenness, thereby reducing reconstruction errors and enhancing forecasting accuracy, as illustrated in Appendix A and Appendix B. CEEMDAN is an extension of the Integrated Empirical Mode Decomposition (EEMD) and the Empirical Mode Decomposition (EMD) algorithms. It effectively addresses the mode overlap issue present in EEMD [27]. The specific steps involved in the CEEMDAN method are outlined below:

Step 1: Data preparation: The input signal is first normalized by dividing it by its standard deviation. This step ensures consistency in the amplitude range of the signal, facilitating more uniform processing in subsequent steps.

Step 2: White noise mode decomposition: Gaussian white noise is added to the signal to generate a new composite signal. EMD is then applied to each iteration of white noise, yielding intrinsic mode functions (IMFs). The first mode is extracted from the decomposed signal, and the residual component is subsequently calculated.

Step 3: Extract residual modes: Residual components are calculated, and the process is repeated using the first-order modal component in place of the original data. This iterative procedure continues until no significant IMF signals remain in the residual. Ultimately, the original signal is fully decomposed into its IMFs.

The primary advantage of this method lies in its ability to account for noise during the signal decomposition process. Additionally, it can adaptively regulate both the noise level and the number of IMFs in the signal.

2.2. The Optimization Algorithm ISSA

In the context of forecasting volatile power prices, selecting key model parameters is crucial before the data is fed into the forecasting model. The ISSA optimization algorithm serves as an effective approach for determining model parameters that minimize the Root Mean Square Error (RMSE) through a global search process. The optimization algorithm builds upon the Sparrow Search Algorithm (SSA) and incorporates an Adaptive Spiral Fly Bird Search Algorithm (ASFSSA) to address challenges such as local optimality and high randomness [28,29]. The ISSA optimization process is outlined as follows:

Step 1: Random variable chaotic diffusion initialization: Population initialization is systematically designed to enhance the algorithm’s controllability.

Step 2: Sparrow food search process: An adaptive weighting is introduced in this process. During the initial stage of the algorithm, this weighting mechanism reduces the impact of random initialization while balancing the subsequent Lévy flight mechanism, thereby enhancing both local and global search capabilities of the algorithm.

Step 3: Lévy flight mechanism: Inspired by the foraging behavior of sparrows, this mechanism demonstrates strong exploratory capabilities, reducing the likelihood of convergence to local optima.

Step 4: Variable spiral search strategy: During the follower position update process, the spiral parameter z cannot remain fixed. This constraint causes the search method to be monotonous and increases the risk of converging to a local optimum, thereby diminishing the algorithm’s search capability. Acting as an adaptive variable, the parameter z dynamically modifies the spiral search trajectory of the follower. This change enables the follower to explore unknown areas more effectively, boosting both search efficiency and the algorithm’s global optimization capability. The variable spiral position update strategy is outlined in the formula below:

Step 5: Iterative optimization: Through continuous iteration, the optimal model parameters are found to make the forecasting effect best

2.3. The Forecasting Model

The forecasting model primarily consists of three modules: CNN, SE Attention and LSTM.

(1) Convolutional Neural Networks

CNNs, inspired by biological processes, are a powerful deep learning tool widely used for classification tasks. The primary components of CNNs include the convolutional layer, pooling layer, and fully connected layer. In time series forecasting, the convolutional layer effectively captures local patterns and features in data, reduces network parameters through the sharing of convolutional kernels, mitigates the risk of overfitting, and enables processing of longer time series data [30].

Specifically, the convolutional layer consists of a rectangular grid of neurons, with its input or preceding layer also arranged in a similar grid. Neurons within a given rectangular region share the same weights, representing the convolutional filter. The pooling layer, in turn, extracts small rectangular sections from the convolutional layer and applies sampling operations, such as average or maximum pooling, to produce a single output for each section. Lastly, the fully connected layer takes all neurons from the previous layers and connects each to every neuron within the layer, enabling the integration of high-level features.

(2) Squeeze-and-Excitation Attention

Originating from image processing, the concept of attention draws inspiration from the human brain’s ability to selectively concentrate on relevant stimuli. By assigning variable weights to features, attention mechanisms emphasize essential information while diminishing the significance of less relevant details [31].

SE Attention, short for Squeeze-and-Excitation, is an attention mechanism designed to enhance CNNs. Its core principle involves incorporating a global attention mechanism to adaptively learn the importance of individual channels [32], as illustrated in Figure 1.

The SE module applies a filter to weight the input data, enhancing the learning of convolutional features. This process improves the network’s sensitivity to informative features, thereby boosting its performance.

(3) Long Short-Term Memory Networks

LSTM networks, equipped with a forget gate, effectively address the challenge of handling continuous time series input [33]. This model has found extensive applications in electricity price forecasting. The LSTM methodology is outlined as follows:

Step 1: Forget gate: The forget gate, a crucial part of LSTM, decides what information to retain or remove from the cell state at time t − 1. The system processes inputs from h_t−1, the previous hidden state, and the current input sequence $x_t$ at time t, yielding an output value ranging from 0 to 1. A value of 1 represents full retention of previous information, whereas a value of 0 signifies complete removal of prior cell state data. This mechanism enables LSTM to effectively manage information flow, reducing the likelihood of losing crucial data in time series analysis.

$$f_t=\sigma (W_f [h_{t-1}, x_t]+b_f)$$

(1)

where $f_t$ represents the state of the forget gate at time t. In the LSTM cell, the forget gate is governed by a weight parameter $W_f$, a bias term $b_f$, and an S-shaped activation function $\sigma$, which collectively regulate the flow of information. These components jointly determine whether to keep or discard data from the prior time step when handling new inputs.

Step 2: Signal Storage: With every time increment, the input gate of the LSTM receives new information $x_t$ and determines which information to retain in the cell state. The hyperbolic tangent layer is then used to calculate the current state of the storage unit $C_t$. By using the intermediate value, the cell state is updated, producing a new cell state.

$$f_t=\sigma (W_f [h_{t-1}, x_t]+b_f)$$

(2)

$$C_t=tanh(Wc [h_{t-1},{ x}_t]+{ b}_c)$$

(3)

$$C_t=f \left(t\right) \otimes C_{t-1}+{ i}_t \otimes C_t$$

(4)

Furthermore, the cell state is parameterized by the weight matrix $Wc$ and the bias term $bc$. The hyperbolic tangent activation function tanh is utilized, along with the Hadamard product $\otimes$.

Step 3: Information Transmission: At time t, the input gate state, denoted as $i_t$, controls the transfer of information from $x_t$ to $C_t$. The associated weight matrix and bias term are represented by $Wi$ and $b_i$, respectively. Furthermore, the cell state is parameterized by the weight matrix $Wc$ and the bias term $bc$. The hyperbolic tangent activation function tanh is utilized, along with the Hadamard product $\otimes$. In an LSTM network, the output gate selects the relevant information from the current cell state to produce the final output. Governed by a sigmoid layer, this gate selects specific segments of the memory cell state for export, and a tanh layer refines the chosen information to produce the output $o_t$.

$$o_t=\sigma (W_o [h_{t-1}, x_t]+b_o)$$

(5)

$$h_t=o_t \otimes tanh C_t$$

(6)

where $h_t$ signifies the hidden state at the current time instance. At time t, the state of the output gate, represented by $o_t$, is calculated using the weight matrix $W_o$ and bias $b_o$.

2.4. Procedures and Frame for the Proposed Model

To accurately predict electricity price data, a novel DROI framework is proposed. The CEEMDAN method is utilized to adaptively decompose time series data into multiple components, which are reconstructed using an econometric t-test to generate data with three distinct frequency states. These three frequency components are processed by CNN to extract trend features, where the CNN channels are weighted by SE Attention, and LSTM is employed to handle time-series dependencies for enhanced forecasting accuracy. To optimize the model’s initial parameters, ISSA is used to find the coefficients that minimize RMSE, thereby improving model interpretability. The overall framework of the proposed method is depicted in Figure 2, with the detailed implementation steps outlined below.

3. Experiment Preparation

This section primarily introduces three key aspects: (1) Reconstruction of the frequency state of the data to be predicted following CEEMDAN decomposition and period division. (2) Benchmarks, evaluation metrics, and network parameters used in the study. (3) Data analysis and decomposition considering economic events and structural breakpoints.

3.1. Data Reconstruction

In the section, correlation analysis and independent sample t-tests are employed to reconstruct the modes into high-frequency, medium-frequency, and low-frequency categories. Modes without correlation to the original sequence (failing the t-test) are designated as high-frequency modes. Modes with correlation but fluctuating around zero are identified as medium-frequency modes. Finally, modes that exhibit correlation and fluctuate around non-zero values (typically the last decomposed modal data) are categorized as low-frequency modes, also referred to as trend components.

After CEEMDAN decomposition, as detailed in Algorithm A1, the electricity price series is separated into 18 components, including 17 $imf$ components and one trend term $res$, as illustrated in Figure 3. The decomposition sequence is arranged in the form of high volatility to low volatility. The high-frequency components, typically representing high-volatility modules, exhibit irregular trajectories, sharp fluctuations, and are centered around a zero mean. These components are influenced by short-term random factors, including temporary policies, transient supply-demand relations, and investment psychology. However, their impact is limited and transient, primarily reflecting random effects. In contrast, mid-frequency components (low-volatility modules) and low-frequency components exhibit trends similar to the original series, reflecting significant factors like market adjustments, impactful policies, or economic cycles. The $res$ component represents the long-term trend of electricity prices, characterized by slow periodic changes. Overall, high-frequency components capture short-term random fluctuations, mid-frequency components reflect policy-induced or cyclical trends with significant impacts, and the low-frequency component (trend term) represents the long-term trend, exerting a decisive influence. As shown in Figure 4, the amplitude of each component increases as the frequency decreases. High-frequency components (random sequences) have a limited impact on the original series, while the influence grows progressively with decreasing frequency.

Based on Figure 3, this section delves into the correlation between each component and the electricity price series, the variance contribution rate of each component, and their average periods. The average period is calculated using the Fast Fourier Transform (FFT) method, with the detailed results presented in

Table 2. Analysis results of imf and trend item res based on CEEMDAN decomposition.

Component	Sample Size	Average Period (Days)	Coefficient of Correlation with the Original Sequence	Variance Contribution Rate (%)
imf1	47,811	0.13	0.0642 **	1.09
imf2	47,811	0.25	0.0901 **	0.09
imf3	47,811	0.50	0.1612 **	1.62
imf4	47,811	0.50	0.2189 **	7.40
imf5	47,811	1.00	0.2162 **	4.93
imf6	47,811	1.00	0.1662 **	2.68
imf7	47,811	2.33	0.1344 **	2.41
imf8	47,811	3.50	0.1600 **	4.98
imf9	47,811	7.00	0.1735 **	7.12
imf10	47,811	15.58	0.1232 **	3.16
imf11	47,811	23.17	0.1384 **	4.65
imf12	47,811	45.29	0.1191 **	2.84
imf13	47,811	104.83	0.1342 **	5.44
imf14	47,811	181.08	0.2590 **	6.93
imf15	47,811	398.42	0.3432 **	18.91
imf16	47,811	664.04	0.3397 **	17.74
imf17	47,811	1992.13	0.2701 **	3.26
res	47,811	1992.13	0.0565	5.44

Note: ** indicates a significance level of 1% and correlation is the Pearson correlation coefficient.

. High-frequency components (shorter periods) exhibit weaker correlations with the original sequence, whereas low-frequency components (longer periods) and the trend component demonstrate stronger correlations with the original series.

As previously analyzed, although the electricity price series exhibits high volatility, the high-frequency components have a relatively minor impact on the original exchange rate compared to their strong periodic characteristics. Thus, these high-frequency components can be aggregated into a single component for analytical forecasting, enhancing the efficiency of subsequent modeling. To identify high-frequency components, we assume that they fluctuate around a zero mean. A one-sample t-test is then conducted to test whether the mean of each decomposed component equals zero, determining their classification as high-frequency components. The test results are presented in

Table 3. t-test of electricity price sequence based on CEEMDAN decomposition.

Component	Sample Size	Detection Value = 0
		T-Value	Degree of Freedom	Significance	95% Confidence Level
					Upper	Floor
imf1	47,811	−0.1475	47,810	0.8827	−0.0000	0.0000
imf2	47,811	−1.4637	47,810	0.1433	−0.0000	0.0000
imf3	47,811	−2.9142	47,810	0.0036	−0.0000	−0.0000
imf4	47,811	−2.0374	47,810	0.0416	−0.0001	−0.0000
imf5	47,811	3.9143	47,810	0.0001	0.0000	0.0001
imf6	47,811	0.7445	47,810	0.4566	−0.0000	0.0000
imf7	47,811	2.2213	47,810	0.0263	0.0000	0.0000
imf8	47,811	3.9277	47,810	0.0001	0.0000	0.0001
imf9	47,811	5.9807	47,810	0.0000	0.0001	0.0001
imf10	47,811	5.5547	47,810	0.0000	0.0000	0.0001
imf11	47,811	7.3254	47,810	0.0000	0.0001	0.0001
imf12	47,811	0.6688	47,810	0.5036	−0.0000	0.0000
imf13	47,811	−9.6685	47,810	0.0000	−0.0002	−0.0001
imf14	47,811	−0.131	47,810	0.0000	−0.0000	0.0000
imf15	47,811	−5.6057	47,810	0.0000	−0.0002	−0.0001
imf16	47,811	32.7985	47,810	0.0000	0.0008	0.0009
imf17	47,811	56.1288	47,810	0.0000	0.0006	0.0006
Res	47,811	8160.712	47,810	0.0000	0.1146	0.1147

, where components with a t-test p-value greater than 0.01 are identified as high-frequency. Consequently, six components $imf1$, $imf2$, $imf4$, $imf6$,$imf7$ and $imf12$ are categorized as high-frequency components. The final decomposition diagram is illustrated in Figure 4.

Furthermore, from

Table 4. The variance contribution rate of each structural component sequence to the electricity price sequence.

Structural Component	Variance Contribution Rate
High frequency component (Random series)	13.65%
Intermediate frequency component (Periodic series)	82.27%
Low-frequency component (Trend series)	4.07%

, similar conclusions can clearly be derived concerning the contribution of each component to the variance of the original sequence. The periodic component contributes the most to the variance, accounting for 82.27%, which highlights its dominant influence on the original series and reflects the highly periodic nature of electricity price fluctuations. Short-period random fluctuations also play a role, with a variance contribution rate of 13.65%. Overall, electricity price fluctuations exhibit an inherent operational trend over the long term, described by the trend component. Medium-frequency components significantly influence electricity prices over shorter periods. Given the relative stability of the Spanish electricity market, the trend component contributes the least to electricity price fluctuations.

3.2. Standard Models, Evaluation Indicators, and Parameter Settings

(1) Standard Models

As the electricity price series is identified as a nonlinear time series, we select only one classical statistical method, ARIMA [34]. Meanwhile, 4 single ML model (LSTM [35], SVR [36], ELM [37] and BP [38]), and 5 hybrid model (LSTM-Attention, RFELM, CCSL, SCSL, CSL) are selected to be benchmark in order to study the forecasting performance of the model. Due to its self-attention mechanism, Transformer can capture data volatility very well. It has good application in electricity price forecasting. Therefore, two hybrid models combined with Transformer (TCN-Transformer-LSTM and ICEEMDAN- TCN-Transformer-LSTM) are selected to be benchmarked in volatile data. To study the robustness of the model. Finally, Naive is introduced as a contrast in the section. Naive is to imitate People’s Daily decisions, according to the electricity price of the previous period, which is directly used as the predicted value of the next period, which can be used to study the economic significance of the model.

(2) Evaluation Indicators

To evaluate the forecasting performance of the model, this study uses two criteria: deterministic forecasting and interval forecasting. Deterministic forecasting offers a straightforward measure of predictive accuracy. Metrics such as mean square error (MSE), mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), and the Lilliefors test are applied. Lower MSE, MAE, RMSE, and MAPE values indicate smaller forecasting errors, demonstrating better model performance. The Lilliefors test examines the goodness-of-fit by testing the normality of residuals. A p-value greater than 0.05 suggests the residual errors are inconsistent with trend information. Interval forecasting, on the other hand, is used to analyze uncertainties in electricity price fluctuations. This study employs interval prediction coverage probability (IPCP), normalized average width of interval forecastings (IPNAW), and cumulative width deviation (AWD) as metrics. A higher IPCP and lower AWD signify the interval captures sufficient true values, improving forecasting reliability. Meanwhile, a smaller IPNAW indicates a narrower forecasting interval and more stable predictive results. The formulas for these calculations are listed in

Table 5. Explanation and formulas for the rules of performance evaluation.

Category	Error	Formula
Ascertainment error	MSE	$E={\displaystyle \frac{\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{n}}$
	RMSE	$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n(y_i-{\widehat{y}}_i{)}^2}{n}}}$
	MAPE	$MAPE={\displaystyle \frac{{\sum }_{i=1}^n|\frac{y_i-{\widehat{y}}_i}{y_i}|}{n}}$
	MAE	$MAE={\displaystyle \frac{{\sum }_{i=1}^n|y_i-{\widehat{y}}_i|}{n}}$
	Lilliefors test	If p is greater than 0.05, there is not enough reason to think that the residuals do not conform to the normal distribution
Interval error [39]	IPCP	$IPCP={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{C}_i$
		$where C_i=\left\{\begin{array}{c}1, if i\in [L_i,U_i]\\ 0, other\end{array}\right.$;
	AWD	$AWD={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{i}_{\alpha }^{AWD}$
		$where i_{\alpha }^{AWD}=\left\{\begin{array}{c}(i_{\alpha }^L-y_i)/(i_{\alpha }^U-i_{\alpha }^L),y_i<i_{\alpha }^L\\ 0,y_i\in i_{\alpha }^L\\ (y_i-i_{\alpha }^U)/(i_{\alpha }^U-i_{\alpha }^L),y_i>i_{\alpha }^U\end{array}\right.$.
	IPNAW	$IPNAW={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{U_i-L_i}{\overline{F}-\underset{\_}{F}}}$

Note: y_i is the actual electricity price, $\widehat{y_i}$ is the forecasting value, n is the sample number, $\alpha$ is the confidence level, $L_i$ is the predicted value of lower bound of interval, $U_i$ is the predicted value of upper bound of interval, $\overline{F}$ is the upper of forecast, $\underset{\_}{F}$ is the lower of forecast.

.

(3) Parameter Settings

This section outlines the key hyperparameters of the hybrid model, as summarized in

Table 6. Hyperparameter settings of the compared models.

Model	Meaning	Value
CSL	Initial learning rate	0.01
	L2 regularization parameter	0.001
	LSTM hidden layer neurons	6
CCSL	Initial learning rate	0.01
	L2 regularization parameter	/
	LSTM hidden layer neurons	6
SCSL	Initial learning rate	ISSA Global Search
	L2 regularization parameter
	LSTM hidden layer neurons
CSCSL	Initial learning rate	ISSA Global Search
	L2 regularization parameter
	LSTM hidden layer neurons

. The learning rate reduction factor for the models is set to 0.05. Additionally, Table 6 lists the learning rate, number of hidden layer nodes, and regularization coefficient for both the SCSL and proposed models, which are optimized through a global search using the improved Sparrow Search Algorithm (ISSA).

3.3. Data Research and Decomposition

Previous research highlights the necessity of adopting a cautious approach when assessing the impact of significant events in various domains [40]. This principle is equally relevant to electricity prices. Forecasting electricity prices over an entire period may fail to effectively capture the intrinsic fluctuations of the electricity market, potentially leading to larger forecasting errors. This study examines the electricity market’s response to two critical events: the fluctuations in the energy market in 2020 and the Russia-Ukraine conflict in 2022. Figure 5 illustrates the trend of electricity price volatility from April 2014 to March 2023.

Affected by energy market volatility, Spanish electricity prices fluctuated in 2020 and achieved explosive growth in 2021. In March 2020, two key factors, namely the rapid increase in natural gas prices and the rising cost of carbon emissions, led to a sharp rise and unprecedented volatility in Spain’s electricity market. By early 2022, these prices had climbed to an all-time high [41]. In February 2022, the Russian-Ukrainian political conflict further exacerbated the situation, driving Spanish electricity prices to extreme levels. Subsequently, electricity prices fluctuated sharply due to two primary factors.

Drawing on the examination of events throughout the study period, the dataset is divided into three distinct phases, with the energy price fluctuations and the Russia-Ukraine conflict serving as discontinuity points, as summarized in

Table 7. Electricity price division chow test.

Discontinuity Point	Dataset	F-Value	p-Value	Conclusion
14 March 2020	Dataset1	109,725.28	0.00	Identify the structural breaks.
	Dataset2
24 February 2022	Dataset2	16,587.28	0.00	Identify the structural breaks.
	Dataset3

. To validate the rationality of this division, the Chow test was employed. The Chow test, a statistical method widely used in time series forecasting, assesses the presence of structural changes. The F-test results for cycle division are presented in Table 6, indicating significant structural shifts both before and after the pandemic and the Russia-Ukraine conflict.

The null hypothesis of the Chow test assumes that the residuals follow an independent and identically distributed normal distribution with unknown variance. Let $S_R$ denote the total sum of squared residuals across all data, $S_1$ the sum of squared residuals for the first period, and $S_2$ the sum of squared residuals for the second period. Here, $N_1$ and $N_2$ represent the number of observations in each period, while $k$ is the total number of parameters. The F-test statistic for the Chow test is calculated as follows:

$$F = {\displaystyle \frac{\left(S_R-\left(S_1+S_2\right)\right)/k}{\left(S_1+S_2\right)/(N_1+N_2-2k)}} ~ F(k,N_1+N_2-2k)$$

(7)

Table 8. Electricity price cycle division.

Dataset	Description	Interval
Dataset1	Before the energy market turbulence	1 April 2014–14 March 2020
Dataset2	The energy price fluctuations	14 March 2020–24 February 2022
Dataset3	The Russia-Ukraine political conflict	24 February 2022–21 March 2023

presents the defined intervals for each period. The period prior to the energy market turbulence (1 April 2014, to 13 March 2020) is characterized as a stable, long-term phase representing typical electricity market behavior. The period of energy price fluctuations (14 March 2020, to 23 February 2022) reflects the significant disruptions caused by the soaring natural gas prices and mounting carbon emission costs. The period of the Russia-Ukraine conflict (24 February 2022, to 21 March 2023) marks a phase influenced primarily by the conflict, compounded by the ongoing effects of the pandemic. As illustrated in Figure 5, during Dataset 1 (Before the coronavirus outbreak), electricity prices remained relatively stable. However, during Dataset 2 (epidemic shock period) and Dataset 3 (Russia-Ukraine conflict period), electricity prices exhibited pronounced fluctuations.

Therefore, we divide the subsequent experiment into two parts: (1) Model forecasting effect experiment: Dataset 1 was used to predict the proposed model without considering the impact of extreme events (without using obvious trend and fluctuation data). (2) Model robustness test: Datasets 2 and 3 were used to study the forecasting effect, and the stability of the model was studied under the scenario with obvious trend influence and high fluctuation. The subsequent experiments are shown in

Table 9. Experiment division table.

Experiment	Goal
Section 4: Model forecasting effect experiment (without large fluctuations and extreme points)	Evaluate model performance
Experiment 1: Model forecasting	Evaluate the forecasting accuracy
Experiment 1.1: Research on basic model forecasting
Experiment 1.2: Research on hybrid model forecasting
Experiment 1.3: Research on interval forecasting
Experiment 2: Ablation experiment	Evaluate the network structure
Experiment 2.1: Decomposition algorithm ablation experiment
Experiment 2.2: Optimization algorithm ablation experiment
Experiment 3: Research on seasonal forecasting	Evaluate the forecasting accuracy
Section 5: Model robustness test (with a large number of volatility cases and extreme points)	Evaluate model stability
Experiment 4: Research on the epidemic cycle	Evaluate the forecasting accuracy
Experiment 5: Research on the Russia-Ukraine conflict cycle	Evaluate the forecasting accuracy
Experiment 6: The DM test	Evaluate model stability

.

4. Model Forecasting Effect Experiment

In this section, a data sample is randomly selected to evaluate the forecasting performance of the model in Dataset 1. To ensure the reliability of subsequent experimental results, the data is split into two subsets in a 7:3 proportion: one for training and the other for testing. To reduce the impact of feature dimension disparities, normalization is applied to the data in this study and subsequent experiments. Normalization, a common preprocessing technique in neural networks, enhances network convergence, mitigates gradient explosion during training, and improves predictive accuracy. The data is scaled using the minimum-maximum normalization method, ensuring values fall between 0 and 1.

4.1. Experiment 1: Model Forecasting

This experiment is divided into three small experiments: (1) comparison with a single model, (2) comparison with a mixed model, and (3) interval forecasting. Through these experiments, deterministic evaluation (MSE, RMSE, MAE, MAPE) and interval evaluation (IPCP, AWD, IPNAW) are applied to evaluate the predictive effect and model stability of the proposed model. The above experiments show that the forecasting accuracy of the proposed model is higher than that of other benchmark models.

4.1.1. Experiment 1.1: Research on Basic Model Forecasting

The effectiveness of the proposed hybrid approach is evaluated against five benchmark models: four machine learning models and one statistical model, as detailed in

Table 10. Research on basic model forecasting.

Model	MSE	RMSE	MAE	MAPE
LSTM	9.16 × 10−6	0.0030	0.0023	2.0606
SVR	1.28 × 10−5	0.0036	0.0027	2.4348
ARIMA	8.60 × 10−6	0.0029	0.0025	2.2573
BP	1.71 × 10−5	0.0041	0.0033	2.9536
ELM	1.44 × 10−5	0.0038	0.0029	2.6173
CNN	1.92 × 10−5	0.0044	0.0029	2.9060
Transformer	2.95 × 10−5	0.0054	0.0041	4.5552
Naive	1.09 × 10−5	0.0033	0.0023	2.0772
The proposed model	3.99 × 10−6	0.0020	0.0015	1.3758

Note: Naive is to imitate the consumption decisions of the public, and take the electricity price of the previous period directly as the predicted value of the next period, which has a strong reference.

.

As shown in Table 10, the proposed model achieves the superior forecasting performance among all benchmarks, with the lowest MAPE of 1.3758%. Nevertheless, it is noteworthy that among the individual base models, LSTM exhibits a marginally better performance than the Naive model. This observation motivated the development of multiple hybrid models to leverage the strengths of LSTM for potentially enhanced forecasting accuracy.

4.1.2. Experiment 1.2: Research on Hybrid Model

To assess the performance of the proposed hybrid method, one single machine learning model and four hybrid models are selected as benchmarks, as shown in

Table 11. Research on Hybrid model.

Model	MSE	RMSE	MAE	MAPE	Lilliefors
LSTM	8.84 × 10−6	0.0030	0.0024	2.1765	0.0074
LSTM-attention	8.93 × 10−6	0.0030	0.0026	2.3644	0.0000
CSL	1.22 × 10−5	0.0035	0.0027	2.3823	0.1426
CCSL	5.78 × 10−6	0.0024	0.0019	1.6839	0.1232
SCSL	1.20 × 10−5	0.0035	0.0026	2.2853	0.0000
The proposed model	3.99 × 10−6	0.0020	0.0015	1.3758	0.3390

Note: 1. Lilliefors normality test: If p is greater than 0.05, there is not enough reason to think that the residuals do not conform to the normal distribution. 2. CCSL represents a CSL model enhanced by CEEMDAN, which performs decomposition and reconstruction of the original sequence. 3.SCSL refers to a CSL model that incorporates ISSA for parameter optimization.4.The proposed model represents the proposed hybrid model that integrates CEEMDAN, ISSA, CNN, SE Attention, and LSTM architectures for enhanced forecasting capability.

. Additionally, an in-depth analysis of the model residuals is conducted, with results presented in

Table 12. Mixed model residual table.

Residuals of Composite Model	Max	Min	Mean	Skewness	Kurtosis
LSTM	0.0105	−0.0082	0.0014	0.0199	4.0061
LSTM-Attention	0.0033	−0.0062	−0.0022	0.5060	2.9786
CSL	0.0108	−0.0101	0.0013	0.0321	3.5058
CCSL	0.0075	−0.0048	0.0015	0.0652	3.7906
SCSL	0.0143	−0.0089	0.0028	0.4378	3.2791
CSCSL	0.0062	−0.0063	0.0002	0.0962	3.5658

Note: Residual is the difference between true value and predicted value.

and illustrated in Figure 6.

Table 11 and Table 12 further demonstrate the forecasting performance and robustness of the proposed model. Among the evaluated models, the proposed model achieves the best predictive accuracy, with residuals following a normal distribution (p-value > 0.05). Additionally, Table 12 and Figure 6 illustrate the residual error distribution, showing that the forecasting errors of the proposed model fluctuate around zero and exhibit relative stability.

4.1.3. Experiment 1.3: Interval Forecasting

To further investigate the model’s predictive stability, its performance in interval forecasting was evaluated against a set of benchmarks models, comprising one machine learning base model and four hybrid models, as presented in

Table 13. Interval forecasting results.

Confidence Interval	Model	IPCP(%)	AWD	IPNAW
95%	LSTM	100.0000	0.0000	0.7419
	SVR	100.0000	0.0000	0.7264
	ARIMA	50.8333	0.3294	0.9567
	BP	100.0000	0.0000	0.6484
	ELM	100.0000	0.0000	0.7258
	RFELM	99.7183	0.0027	0.6984
	LSTM-Attention	100.0000	0.0000	0.6780
	CSL	100.0000	0.0000	0.7496
	CCSL	100.0000	0.0000	0.7891
	The proposed model	100.0000	0.0000	0.7387
90%	LSTM	100.0000	0.0000	0.6226
	SVR	99.7183	0.0024	0.6096
	ARIMA	43.8889	0.5318	0.8029
	BP	98.5915	0.0118	0.5442
	ELM	100.0000	0.0000	0.6091
	RFELM	99.7183	0.0027	0.5861
	LSTM-Attention	100.0000	0.0000	0.5690
	CSL	100.0000	0.0000	0.6291
	CCSL	100.0000	0.0000	0.6622
	The proposed model	100.0000	0.0000	0.6199
85%	LSTM	100.0000	0.0000	0.5449
	SVR	99.1549	0.0069	0.5335
	ARIMA	40.0000	0.5626	0.7026
	BP	97.7465	0.0159	0.4762
	ELM	99.4366	0.0043	0.5330
	RFELM	98.8732	0.0086	0.5129
	LSTM-Attention	99.4805	0.0040	0.4980
	CSL	100.0000	0.0000	0.5505
	CCSL	100.0000	0.0000	0.5796
	The proposed model	100.0000	0.0000	0.5426
80%	LSTM	99.7183	0.0020	0.4845
	SVR	97.4648	0.0188	0.4750
	ARIMA	37.7778	0.5833	0.6255
	BP	96.9014	0.0243	0.4240
	ELM	99.1549	0.0063	0.4745
	RFELM	98.0282	0.0153	0.4567
	LSTM-Attention	98.9610	0.0077	0.4433
	CSL	100.0000	0.0000	0.4901
	CCSL	100.0000	0.0000	0.5160
	The proposed model	100.0000	0.0000	0.4830
75%	LSTM	99.7183	0.0021	0.4349
	SVR	95.2113	0.0319	0.4263
	ARIMA	35.5556	0.6052	0.5615
	BP	94.6479	0.0384	0.3806
	ELM	97.7465	0.0150	0.4260
	RFELM	96.3380	0.0263	0.4099
	LSTM-Attention	98.1818	0.0132	0.3979
	CSL	100.0000	0.0000	0.4399
	CCSL	100.0000	0.0000	0.4631
	The proposed model	100.0000	0.0000	0.4336

. The comparative results are visually summarized in Figure 7.

Table 13 presents the interval forecasting performance. Interval forecasting provides varying confidence levels for the predicted results, offering insights into their reliability and robustness. At confidence levels below 80%, only CSL, CCSL, and the proposed model achieve complete coverage of true values (IPCP = 100%). Among these models, the proposed model exhibits the narrowest interval range (IPNAW = 0.4336 at a 75% confidence level and IPNAW = 0.4830 at an 80% confidence level), demonstrating the highest predictive accuracy. Additionally, the results indicate that single ARIMA model is unsuitable for electricity price forecasting.

Remark: In the first experiment, interval and deterministic evaluations are combined, demonstrating the proposed model’s strong forecasting performance and stability. Therefore, subsequent experiments focus on analyzing the contributions of each module within the proposed model.

4.2. Experiment 2: Ablation Experiment

This experiment consists of two sub-experiments: (1) evaluation of the decomposition module’s performance; and (2) evaluation of the optimization module’s performance. Deterministic metrics (MSE, RMSE, MAE, MAPE) are employed to assess the predictive accuracy and stability of the proposed model. The experimental results clearly demonstrate the positive impact of the decomposition and optimization modules on the performance of the proposed model.

4.2.1. Experiment 2.1: Decomposition Algorithm Ablation Experiment

To evaluate the impact of the decomposition module, an ablation experiment focusing on decomposition (including the reconstruction process) was conducted, as presented in

Table 14. Optimization effect study of CEEMDAN.

Model	MSE	RMSE	MAE	MAPE
ISSA-CNN-SE Attention-LSTM	1.20 × 10−5	0.0035	0.0026	2.2853
ICEEMDAN-ISSA-CNN-SE Attention-LSTM	9.43 × 10−6	0.0031	0.0022	1.9877
The proposed model	3.99 × 10−6	0.0020	0.0015	1.3758

Note: The optimal parameters obtained by ISSA optimization algorithm are as follows: the learning rate is 0.0247, the number of hidden layer neurons is 6, the regularization coefficient is 0.0001, and the corresponding fitness (RMSE) is 0.0042.

.

Table 14 shows the optimization effect of CEEMDAN decomposition and reconstruction. The results show that MAPE of the proposed model is reduced from 2.96641% to 1.3758% and RMSE from 0.0046 to 0.0020. Furthermore, ICEEMDAN decomposition and reconstruction method is introduced for comparison. ICEEMDAN increases the setting of the signal-to-noise ratio compared with the CEEMDAN algorithm to standardize the noise. However, with the addition of signal-to-noise ratio, the forecasting effect is not significantly improved. This may be due to the following reasons: Electricity price data are highly volatile due to market supply and demand. This volatility poses two risks to the Increased Signal-to-Noise Ratio (SNR) module: (1) It may erroneously filter out subtle, meaningful fluctuations stemming from societal shifts in user behavior, mistaking them for noise, thereby discarding critical features of the original signal. (2) It may be susceptible to rare events, which can introduce anomalous errors that distort the signal’s inherent characteristics. Both scenarios can compromise the model’s forecasting performance by altering the fundamental data upon which it relies.

4.2.2. Experiment 2.2: Optimization Algorithm Ablation Experiment

To evaluate the impact of the ISSA optimization module, an ablation experiment focusing on optimization was conducted, as presented in

Table 15. Compares the optimization effect of ISSA.

Model	MSE	RMSE	MAE	MAPE
CNN-SE Attention-LSTM	1.22 × 10−5	0.0035	0.0027	2.3823
CEEMDAN-CNN-SE Attention-LSTM	5.78 × 10−6	0.0024	0.0019	1.6839
The proposed model	3.99 × 10−6	0.0020	0.0015	1.3758

.

Table 15 presents the optimization effect of the ISSA algorithm. The experimental results indicate that the forecasting performance of the proposed model improves when its parameters are optimized using the ISSA algorithm, with MAPE reducing from 1.6839% to 1.3758%.

Remark: The results of the above experiments indicate that both the CEEMDAN decomposition and reconstruction algorithm and the ISSA parameter optimization algorithm enhance the forecasting performance of the model. Notably, the combination of these two methods (decomposing the data first and then optimizing parameters for each frequency state) significantly improves forecasting accuracy. Specifically, MAPE decreased from 2.3823% to 1.3758%, as illustrated in Table 15.

4.2.3. Experiment 3: Research on Seasonal Forecasting

Based on the previous experiments, Spain, located in southern Europe and surrounded on three sides by the Mediterranean Sea, is influenced by warm, moist air from the Mediterranean and monsoons from the North Atlantic. These climatic conditions create distinct seasonal patterns. To explore the predictive enhancements provided by each module in the proposed model, seasonal forecasting is introduced. The experiment evaluates performance improvements across various seasons using interval and deterministic forecastings, as presented in

Table 16. Season interval forecasting hybrid model points.

Confidence Level (%)	Model	Spring			Summer
		IPCP(%)	AWD	IPNAW	IPCP(%)	AWD	IPNAW
95	CSL	99.5772	0.0039	0.6887	99.7886	0.0020	0.8801
	CCSL	100.0000	0.0000	0.6792	100.0000	0.0000	0.8673
	SCSL	100.0000	0.0000	0.7329	99.7886	0.0020	0.9101
	The proposed model	100.0000	0.0000	0.6788	100.0000	0.0000	0.8538
90	CSL	98.9429	0.0091	0.5780	99.5772	0.0038	0.7386
	CCSL	100.0000	0.0000	0.5697	99.7886	0.0018	0.7279
	SCSL	99.7886	0.0017	0.6151	99.7886	0.0019	0.7638
	The proposed model	100.0000	0.0000	0.6230	100.0000	0.0000	0.7166
85	CSL	97.0402	0.0231	0.5058	98.0973	0.0147	0.6464
	CCSL	99.7886	0.0014	0.4985	99.7886	0.0018	0.6370
	SCSL	99.1543	0.0063	0.5383	99.1543	0.0065	0.6685
	The proposed model	100.0000	0.0000	0.5452	100.0000	0.0000	0.6271
80	CSL	95.9831	0.0317	0.4503	96.4059	0.0274	0.5755
	CCSL	99.7886	0.0016	0.4438	99.7886	0.0018	0.5671
	SCSL	99.1543	0.0067	0.4792	98.5201	0.0113	0.5951
	The proposed model	100.0000	0.0000	0.4854	100.0000	0.0000	0.5583
75	CSL	94.2918	0.0427	0.4042	95.1374	0.0355	0.5166
	CCSL	99.5772	0.0028	0.3984	99.5772	0.0032	0.5091
	SCSL	97.4630	0.0174	0.4302	96.6173	0.0234	0.5342
	The proposed model	99.7886	0.0011	0.4357	100.0000	0.0000	0.5011
Confidence Level (%)	Model	Autumn			Winter
		IPCP(%)	AWD	IPNAW	IPCP(%)	AWD	IPNAW
95	CSL	99.5772	0.0040	0.6802	100.0000	0.0000	0.8558
	CCSL	100.0000	0.0000	0.6179	100.0000	0.0000	0.8235
	SCSL	99.7886	0.0020	0.7829	100.0000	0.0000	0.8545
	The proposed model	100.0000	0.0000	0.6518	100.0000	0.0000	0.8063
90	CSL	99.3658	0.0057	0.5708	99.7886	0.0018	0.7182
	CCSL	100.0000	0.0000	0.5185	100.0000	0.0000	0.6911
	SCSL	99.7886	0.0020	0.6570	100.0000	0.0000	0.7171
	The proposed model	99.7908	0.0018	0.5470	100.0000	0.0000	0.6767
85	CSL	98.5201	0.0120	0.4996	99.7886	0.0018	0.6286
	CCSL	100.0000	0.0000	0.4538	100.0000	0.0000	0.6048
	SCSL	99.5772	0.0036	0.5750	99.7886	0.0017	0.6276
	The proposed model	99.7908	0.0018	0.4787	100.0000	0.0000	0.5922
80	CSL	98.0973	0.0155	0.4447	99.3658	0.0046	0.5596
	CCSL	99.5772	0.0028	0.4040	100.0000	0.0000	0.5384
	SCSL	99.1543	0.0064	0.5119	99.5772	0.0033	0.5587
	The proposed model	99.7908	0.0018	0.4262	100.0000	0.0000	0.5272
75	CSL	97.2516	0.0207	0.3992	98.7315	0.0088	0.5023
	CCSL	99.5772	0.0031	0.3626	100.0000	0.0000	0.4833
	SCSL	98.3087	0.0119	0.4595	99.5772	0.0034	0.5015
	The proposed model	99.5816	0.0032	0.3826	100.0000	0.0000	0.4732

and

Table 17. Hybrid model seasonal forecasting.

Season	Model	MSE	MAPE	RMSE	MAE
Spring	CSL	4.31 × 10−5	2.8427	0.0049	0.0036
	CCSL	9.67 × 10−6	1.9496	0.0031	0.0024
	SCSL	1.58 × 10−5	2.6254	0.0043	0.0033
	The proposed model	6.31 × 10−6	1.5419	0.0025	0.0019
Summer	CSL	1.77 × 10−5	2.5673	0.0042	0.0030
	CCSL	7.80 × 10−6	1.8245	0.0028	0.0022
	SCSL	1.87 × 10−5	2.5380	0.0040	0.0029
	The proposed model	5.02 × 10−6	1.5147	0.0022	0.0018
Autumn	CSL	4.53 × 10−5	3.8948	0.0067	0.0050
	CCSL	2.26 × 10−5	2.9450	0.0048	0.0037
	SCSL	2.96 × 10−5	3.0397	0.0054	0.0039
	The proposed model	2.47 × 10−5	2.8635	0.0050	0.0037
Winter	CSL	2.42 × 10−5	4.4310	0.0066	0.0051
	CCSL	2.24 × 10−5	3.7247	0.0047	0.0040
	SCSL	3.58 × 10−5	3.7477	0.0060	0.0045
	The proposed model	6.52 × 10−6	1.6571	0.0026	0.0019

.

Table 16 presents the results of ablation experiments for seasonal interval forecasting. Compared to other hybrid models, the proposed model achieves the highest interval coverage and the narrowest interval width. Notably, during the Spanish summer, electricity price volatility increases significantly due to heat waves. At the 90% confidence level and below, only the proposed model achieves full coverage of true values (IPCP = 100%). Additionally, across all seasons, CSL consistently exhibits the lowest interval coverage. Both ISSA and CEEMDAN contribute to optimizing the model’s performance to varying degrees. Furthermore, a seasonal deterministic point experiment is conducted to further investigate the module effects within the proposed model.

Table 17 presents the results of ablation experiments for seasonal point forecasting. In every season, the proposed model achieves the lowest forecasting error. Regarding the ISSA and CEEMDAN optimization modules: (1) Both modules enhance the forecasting performance of CSL across all seasons; and (2) CEEMDAN demonstrates significant improvement in forecasting accuracy during spring and summer. This is attributed to its ability to perform local fitting for varying data frequency states, effectively capturing local fluctuations induced by seasonal factors, such as minor heat waves.

Remark: The ISSA multi-objective optimization algorithm is crucial for determining key hyperparameters of the model, including the number of hidden layer nodes, learning rate, and regularization coefficient. By optimizing these parameters, ISSA effectively reduces forecasting errors caused by suboptimal parameter settings. Meanwhile, CEEMDAN enhances CNN’s ability to extract critical trend information through wave frequency decomposition. Combined with the Attention mechanism, this approach improves both the model’s predictive accuracy and the effectiveness of interval coverage.

5. Model Robustness Test

To validate the robustness of the proposed CSCSL system, it is applied to electricity price data from the epidemic cycle and the Russia-Ukraine conflict cycle. Given the high volatility of the forecasts, interval forecasting is utilized in this experiment. Additionally, TCN and transformer mechanisms, known for their strong spatial information extraction capabilities, are incorporated. For this study, TCN-Transformer-LSTM (TCL) and ICEEMDAN-TCL (ITCL) are introduced, corresponding to Experiment 4 and Experiment 5, respectively.

5.1. Experiment 4: Research on the Epidemic Cycle

This experiment conducted a comprehensive forecasting for the energy price fluctuation period. During this time, the Spanish electricity market exhibits a general upward trend with pronounced volatility, driven by the pandemic and other factors. To evaluate the robustness of the research model, periodic data are utilized. In the experiment, the training and validation sets are divided in a 7:3 ratio, as depicted in Figure 8.

Table 18. Forecasting data during the epidemic cycle.

Model	CSL	CSCSL	TTL	ITTL	SVR	RFELM
IPCP(%)	87.0847	89.1374	75.7792	57.1348	58.8810	80.4919
AWD	0.1235	0.1040	0.2348	0.4091	0.4011	0.1895
IPNAW	0.6889	0.4684	0.6274	0.6508	0.4181	0.5093

Note: The 95% confidence level was used for interval forecasting.

summarizes the interval error for each model during this period. At a 95% confidence level, the forecasting interval coverage of SCL and CSCSL reaches approximately 90%, significantly outperforming other models.

5.2. Experiment 5: Research on the Russia-Ukraine Conflict Cycle

This experiment conducted a comprehensive forecasting for the Russia-Ukraine political conflict period. During this time, energy prices across the EU experienced significant fluctuations due to political and geopolitical tensions, and Spanish electricity prices followed a similar trend. To evaluate the robustness of the research model, periodic data were utilized. The training and validation sets were divided in a 7:3 ratio, as depicted in Figure 9.

Table 19. Forecasting data during Russia–Ukraine conflict period.

Model	CSL	CSCSL	TTL	ITTL	SVR	RFELM
IPCP(%)	99.9581	100.0000	99.7902	99.0348	99.8320	98.6140
AWD	0.0004	0.0000	0.0017	0.0063	0.0012	0.0094
IPNAW	0.7755	0.6311	0.4531	0.3663	0.4079	0.3624

Note: The 95% confidence level was used for interval forecasting.

presents the interval error for each model during this period. The CSCSL model achieved a forecasting interval coverage of 100% at a 95% confidence level, indicating that all true values were contained within the predicted confidence interval. The CSL model also performed well, with an IPCP coverage rate of 99.9581%. While the proposed model demonstrated stronger forecasting performance, its forecasting interval was wider compared to other models.

5.3. Experiment 6: The Diebold-Mariano (DM) Test

In this section, the DM test is introduced, and the sample error t-test is used to check whether the forecasting efficiency of the two models is the same. By predicting, we receive the predicted error sequence $E_A$ and $E_B$. $E_A$ and $E_B$ are the true values (before data processing) of the two models minus the predicted values.

$$\overline{d}={\displaystyle \frac{E_A-E_B}{n}}$$

(8)

$$DM={\displaystyle \frac{E_A-E_B}{\sqrt{\frac{\sum _{i=1}^n\left(\right(E_A-E_B)-\overline{d})}{n-1}}}}$$

(9)

If the Diebold-Mariano (DM) statistic follows a normal distribution, it suggests no significant difference between the two models. Otherwise, model performance can be evaluated based on the study framework, as illustrated in Figure 10 and Figure 11. Figure 10 depicts the forecasting performance of each model during the epidemic cycle, while Figure 11 illustrates the results for the Russia-Ukraine war cycle. A negative DM value indicates that model B has a smaller error compared to model A, suggesting that model A demonstrates superior forecasting performance.

Combined with Experiments 4 and 5, the proposed model demonstrates considerable robustness to the prolonged effects of sudden events. This is particularly evident in electricity markets affected by long-term emergencies, such as the epidemic shock, which exhibit both high volatility and pronounced long-term trends. The proposed model leverages feature decomposition (CEEMDAN) to extract trend components from such data and conducts predictive research separately within distinct modules. As indicated by the interval forecasting results and the DM test, the forecasting error of the proposed model is significantly smaller compared to other models (Table 18 and Figure 10).

However, for unexpected events with long-term market impacts, the proposed model achieves significantly higher interval forecasting accuracy compared to other models, albeit with a larger forecasting bandwidth. In such cases, combining the proposed model with other models (e.g., TTL, SVR) can be considered to enhance forecasting accuracy (Table 19 and Figure 11).

6. Conclusions

Accurate electricity price forecasting is critical in highly competitive markets for plant operators, market operators, and participants. This study focuses on addressing the challenges of short-term electricity price forecasting, using the Spanish electricity market as a case study. The market data is divided into multiple subsets based on economic events and breakpoint tests. Decomposition and reconstruction techniques are then employed to segment the data into modal datasets, with ISSA determining the optimal parameters. For each frequency state dataset, CNN is utilized for feature extraction, enhanced by SE Attention for channel weighting, and LSTM is applied for time series forecasting. Compared with other comparative methods, the average Mean Absolute Percentage Error (MAPE) of this method is reduced by 46.01%. This can effectively help electricity retailers and industrial users alleviate the “cost mismatch risk” while assisting power generation companies in optimizing resource allocation, thereby enhancing the overall profit level of the industry.

The benchmark suite in this study comprises one econometric model, six machine learning models, seven hybrid models, and naive models designed to simulate daily decision-making. The forecasting performance and robustness of the proposed model are evaluated using both fluctuating and non-fluctuating datasets. Additionally, the study accounts for seasonal factors and performs a detailed analysis using seasonal models. The experimental results reveal that: (1) the proposed model achieves superior forecasting performance compared to other benchmarks; (2) the decomposition and reconstruction modules contribute to significant model optimization; (3) the proposed model outperforms Naive methods and demonstrates promising application potential. Future research could focus on: (1) Performance of economic analyses on each decomposed module to assess the impact of major events or economic policies; and (2) Integration of error correction mechanisms for datasets with pronounced volatility to improve forecasting accuracy.

However, this study has several potential limitations that require further exploration. First, the case study is confined to the Spanish electricity market, and its applicability to other markets with different characteristics (e.g., varying regulatory frameworks or energy mix structures) remains untested. Second, while the study accounted for the impact of economic events on electricity prices and conducted breakpoint tests, it did not delve into the underlying political and economic factors or the specific mechanisms through which they exert influence.

Building on the aforementioned limitations and findings, future research can be advanced in the following aspects: (1) Expand the data scope to include multi-regional and multi-type electricity markets, and incorporate cross-market comparative analysis to enhance the model’s generalization ability. (2) Integrate econometric analysis into the decomposition module, and introduce dummy variables to quantify the impact of major economic events (such as carbon policy adjustments and international natural gas price shocks) on each modal dataset. This will help establish linkages between economic mechanisms and model features, thereby improving the interpretability of forecasting results.