The World Meteorological Organization (WMO) defines a heatwave as a period during which the daily maximum temperature remains above 32 °C for at least three consecutive days. However, considering that a fixed temperature threshold is difficult to generalize across large spatial scales with varying climatic conditions, the relative threshold method is often more effective for characterizing regional high temperature events. Following the approach in reference (Xu et al., 2016), this study establishes a regional high temperature criterion: if the daily maximum temperature T _max,a on day a exceeds the 98th percentile value T ₉₈ of the historical temperature series for that region, day a is classified as a regional high temperature day. Furthermore, when such high temperature days persist for three or more consecutive days, the event is defined as a regional heatwave.

To identify heatwave days, a heatwave event index H _wa is introduced to determine whether day D _H constitutes a heatwave day. When H _wa >0, day D _H is recognized as a heatwave day. The specific calculation of the heatwave event index H _wa is given in Equation 1: H w a = 1 , if   ∑ i = 0 n − 1 I T max ⁡ , a + i > T 98 ≥ 3 0 , else (1)

In the equation, H _wa denotes the heatwave event index used to determine whether day a belongs to a heatwave period; I (*) is an indicator function that equals 1 if the condition inside the parentheses holds true, and 0 otherwise; n represents the length of the consecutive time window (e.g., n = 3 indicates a 3-day continuous period). When H _wa >0, day a is identified as a heatwave day.

Bidirectional Gated Recurrent Unit (BiGRU) (Niu et al., 2022) is a widely used recurrent neural network (RNN) model in time series forecasting and natural language processing. It captures contextual information of sequential data more effectively by combining forward and backward GRU layers.

The gated recurrent unit (GRU) is an optimized variant of the traditional RNN, engineered to alleviate the vanishing gradient problem through its specialized gating mechanism. It consists of two core components—the reset gate and the update gate—that control the flow of information within the network.

In contrast to the long short-term memory (LSTM) unit, the GRU boasts a more streamlined structure, employing only these two gates to regulate information transmission. This simplified design enhances computational efficiency while retaining predictive performance on par with LSTMs. The architectural framework of the GRU is illustrated in Figure 1.

The BiGRU builds upon the standard GRU by integrating two parallel GRU layers: one processes the input sequence in a forward direction (from past to future), while the other operates in a backward direction (from future to past). The outputs of these two directional layers are concatenated to capture more comprehensive contextual information.

Unlike unidirectional GRUs, the BiGRU simultaneously leverages hidden states from both the forward and backward passes, enabling it to model more holistic temporal dependencies. This bidirectional design proves especially effective for datasets where both past and future contextual information collectively influence the current state. The architectural structure of the BiGRU is illustrated in Figure 2.

The Sequence-to-Sequence (Seq2Seq) architecture is widely employed for modeling sequential data and has demonstrated strong performance in tasks such as machine translation, text generation, and time series forecasting. A standard Seq2Seq model comprises two core components: an encoder and a decoder, which are typically implemented using RNN, LSTM networks, or GRU.

The attention mechanism (Vaswani et al., 2017) stands as a critical technique in deep learning, significantly enhancing the modeling of sequential data. Within a traditional Seq2Seq framework, the encoder condenses the entire input sequence into a fixed context vector c, and the decoder generates outputs based solely on this singular vector. However, for lengthy input sequences, this static representation often struggles to encapsulate all relevant information, restricting the model’s ability to effectively capture long-range dependencies.

To address this constraint, the attention mechanism empowers the decoder to dynamically focus on different segments of the input sequence at each time step t, rather than relying exclusively on a single context vector. By assigning a weight to each position in the input sequence, attention quantifies the relevance of each element to the current decoding step—thereby strengthening the model’s capacity to capture long-range dependencies. This process is formally defined in Equation 6: Attention Q , K , V = softmax Q K ⊤ d k V (6)

In this equation, Q, K, and V represent the query, key, and value matrices respectively, with d_k denoting the dimension of the key vector and n signifying the sequence length. However, the traditional attention mechanism exhibits computational and memory complexity of O (n²), which grows quadratically with sequence length. This creates substantial challenges for efficiency and scalability when handling long sequences.

To overcome this limitation, this study introduces a Dilated Sliding Window (DSW) attention mechanism (as shown in Figure 3). This mechanism expands the receptive field of self-attention, allowing the model to more effectively capture long-range dependencies while preserving computational efficiency—ultimately enhancing the overall forecasting performance.

The attention mechanism improved with the DSW approach is formulated as Equation 7: DilatedLocalAttention i = Attention   q i , K i − s · d : s · d : i + s · d , V i − s · d : s · d : i + s · d (7)

In this equation, q _i represents the query vector of the i-th token; s denotes the half-width of the sliding window (step count); d is the dilation rate, which specifies the interval of skipped elements. K i − s · d : s · d : i + s · d denotes selecting keys from positions i−sd to i + sd at intervals of d., and correspondingly, the values V are selected from the same positions.

The Inception network (Szegedy et al., 2015), proposed by Google, is an improved convolutional neural network designed to enhance representational capability while maintaining computational efficiency. Its core idea is to extract features through the parallel operation of convolution kernels with multiple receptive field sizes, enabling the effective fusion of information from different scales and capturing multi-scale feature patterns.

When used as a feature extractor, the Inception network stacks parallel convolutional and pooling layers to achieve efficient and expressive feature representations. In the Seq2Seq architecture, the subsequent BiGRU network models and predicts the temporal features extracted by the Inception module, further improving the accuracy of sequence forecasting.

The Inception network structure used in this paper is illustrated in Figure 4.

The Grey Wolf Optimizer (GWO) (Mirjalili et al., 2014) is a metaheuristic algorithm that draws inspiration from the hierarchical social structure and collaborative hunting behaviors of grey wolves. Within the algorithm, the population is divided into four distinct tiers: alpha (α) wolves, which correspond to the current optimal solution; beta (β) and delta (δ) wolves, which support the alpha in guiding the search process; and omega (ω) wolves, which make up the rest of the population. By simulating the grey wolves’ natural tactics of encircling and chasing prey, the algorithm effectively steers the search toward globally optimal solutions.

The encircling behavior is modeled mathematically as shown in Equations 8, 9: D → = C → · X → p − X → (8) X → t + 1 = X → p − A → · D → (9) where X → p is the position of the prey (best solution), X → is the current position, A → = 2 · a → · r → 1 − a → , C → = 2 · r → 2 , and a → linearly decreases from 2 to 0 over iterations to balance exploration and exploitation.

Each wolf updates its position based on the three best solutions (α, β, δ), as shown in Equations 10, 11: X → t + 1 = X → 1 + X → 2 + X → 3 3 (10) X → i = X → leader i − A → i · C → i · X → leader i − X → (11)

This mechanism ensures convergence towards the optimal solution while maintaining population diversity.

GWO is simple, requires no gradient information, and effectively balances exploration and exploitation, making it suitable for continuous and complex optimization problems.

To address the challenge of insufficient prediction accuracy in user-level short-term load forecasting—particularly under extreme weather conditions such as high-temperature heatwaves—this paper proposes a novel Attention-enhanced Inception-BiGRU-Seq2Seq (AIBS) model. The overall architecture of the proposed model is illustrated in Figure 5 and comprises four key functional components, each meticulously designed to enhance feature representation, sequence modeling, attention allocation, and parameter optimization.

Inception-based Feature Extraction Layer: To comprehensively capture the diverse temporal patterns and external influencing factors affecting electricity load—such as weather variables (e.g., temperature, humidity, wind speed) and historical consumption data—the model employs an Inception-based module as the initial feature extractor. This structure integrates multiple convolutional kernels with varying receptive fields operating in parallel, enabling simultaneous extraction of finegrained local features and broader contextual information. Such multi-scale representation significantly enriches the feature space and bolsters the model’s capability to characterize the complex, nonlinear relationships between load profiles and meteorological factors, especially pronounced during extreme weather events.

The overall forecasting workflow comprises five key steps, each carefully designed to ensure data integrity, feature relevance, model robustness, and thorough performance evaluation. The complete process is illustrated in Figure 6.

Data Preprocessing and Normalization: The raw user-level load data first undergo comprehensive preprocessing to enhance data quality and suitability for model training. This stage includes several operations: outlier detection and removal are applied to eliminate anomalous readings caused by sensor malfunctions or transmission errors, thereby ensuring the training data accurately reflect typical load behaviors. Missing values—often resulting from equipment downtime or recording failures—are addressed using appropriate imputation methods such as linear interpolation or nearest neighbor techniques, maintaining temporal continuity. Subsequently, normalization methods, typically min-max scaling or z-score standardization, are applied to all input features. This step mitigates discrepancies due to varying scales and units across features, stabilizes gradient descent during training, accelerates convergence, and ultimately improves both learning efficiency and model stability.

To evaluate the effectiveness of the proposed AIBS model, real-world electricity load data were collected from multiple user categories within a specific region. This dataset covers the period from January 1, 2015, to December 31, 2022, with a sampling interval of 15 min. Beyond load measurements, the dataset also incorporates meteorological variables that are closely linked to electricity consumption—including temperature and humidity. These variables support in-depth analysis of how weather conditions influence fluctuations in electricity load.

The AIBS model adopts a direct multi-step forecasting approach: it takes 72 h of historical electricity load data and corresponding meteorological data as input, and uses this information to predict the electricity load for the subsequent 24 h. We divided the dataset into training, validation, and test sets in a 6:3:1 ratio.

A detailed summary of the dataset’s contents is presented in Table 1.

Figure 7 illustrates the temporal evolution of high-temperature days and heatwave days from 2015 to 2022.

As shown, both the frequency of high-temperature events and the occurrence of heatwave episodes exhibit an overall upward trend during the observation period. The annual number of high-temperature days shows a clear increasing trajectory, reflecting the gradual intensification of extreme heat conditions. Similarly, the number of heatwave days also demonstrates a notable rise. These consistent increases indicate that extreme thermal events are becoming both more frequent and more prolonged, posing significant challenges to power system operation and short-term load forecasting. The observed trends highlight the necessity of developing forecasting models capable of effectively capturing load variations under increasingly extreme climatic conditions.

The proposed model was implemented in a Python 3.11 environment, with the PyTorch deep learning framework utilized for both model development and training. Experimental trials were carried out on a hardware setup featuring a 14th-generation Intel Core i7 processor, 32 GB of RAM, and an NVIDIA RTX 4090 GPU. This configuration delivers robust computational capabilities, guaranteeing efficient and stable execution of both training iterations and prediction tasks.

This study employs Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) as quantitative evaluation metrics to assess the predictive performance of the models.

RMSE measures the deviation between predicted and actual observed values, and its calculation is expressed in Equation 12: R M S E = ∑ i = 1 n Y ^ i − Y i 2 n (12)

MAE reflects the average magnitude of the absolute errors between predicted values and actual observations. Its calculation is expressed in Equation 13: M A E = 1 n ∑ i = 1 n y ^ i − y i (13)

In the equations, y _i denotes the actual observed value at time i; y ^ i represents the predicted value at time i; and n is the total number of sampling points.

The closer the MAE value is to zero, the smaller the error between the predicted and actual values, indicating better model fitting performance. When the predicted values perfectly match the true values, the MAE equals zero. MAE provides an intuitive measure of error magnitude—the larger the value, the poorer the model’s predictive accuracy.

After selecting the dataset under heatwave conditions using the relative threshold method, this study further analyzes the correlation between multiple users’ load data and features such as meteorological factors and electricity prices using the Maximal Information Coefficient (MIC) algorithm. The correlation results among features are presented in the heatmap shown in Figure 8, which intuitively illustrates the degree of influence each feature has on load variations.

The results of the correlation analysis reveal that, under heatwave conditions, different categories of user loads exhibit distinct sensitivities to meteorological and external features. Among all considered variables, temperature-related features demonstrate the strongest overall influence, with air temperature and surface temperature emerging as the most critical factors, followed by dew point temperature. This is consistent with the physical mechanism whereby elevated ambient temperatures directly drive the increased use of air conditioning and cooling systems, resulting in sharp rises in electricity demand.

In addition to temperature variables, solar irradiance shows a moderate but notable correlation with load, primarily due to its strong physical relationship with surface temperature and its indirect influence on indoor thermal comfort requirements. Relative humidity also plays an important role, especially in environments such as shopping malls, residential communities, and technology companies, where both indoor climate control systems and occupant comfort levels are highly sensitive to humidity variations.

Interestingly, for certain industrial loads, such as sand and gravel plants, the analysis indicates that electricity prices exert a dominant influence on load variation. This reflects the operational strategy of such facilities, which often schedule high-energy-consuming processes during off-peak nighttime hours to minimize electricity costs, thereby creating a load profile strongly tied to tariff structures rather than weather conditions.

Based on these findings, a feature selection strategy is formulated to ensure that only highly relevant and physically meaningful variables are included as model inputs, thereby reducing redundancy and improving training efficiency. The resulting selected input features for each user load category are summarized in Table 2, serving as the foundation for the subsequent model development and forecasting experiments.

Given the complexity of the proposed AIBS model structure, which contains multiple critical hyperparameters in its encoder and decoder networks, this study employs Grid Search, Random Search, and the proposed GWO algorithm for hyperparameter optimization.

While Grid Search offers exhaustive coverage, its computational cost is extremely high, with the complete evaluation process taking 1,021.83 min. Random Search has the lowest time cost but, due to its inherent randomness, often struggles to find the optimal hyperparameter combination, adversely affecting model performance.

In contrast, the GWO optimization strategy proposed in this paper reduces the hyperparameter tuning time significantly to 463.96 min. Within fewer than 200 iterations, this method identifies stable and reliable hyperparameter settings within an acceptable timeframe, simultaneously ensuring prediction accuracy and greatly improving computational efficiency.

It is noteworthy that GWO effectively alleviates the burden of manual tuning, enhances the model’s generalization capability, and reduces the risk of overfitting.

The hyperparameter optimization times of each method are summarized in Table 3.

To ensure the fairness and reliability of the experimental results, the hyperparameters of the proposed model were carefully configured, with detailed settings listed in Table 4. Additionally, to achieve fair comparisons, all baseline models were also subjected to hyperparameter tuning to guarantee that each model participated in the experiments under its respective optimal configuration.

This consistent and reasonable hyperparameter setting and optimization strategy ensure that all models are compared under identical experimental conditions, yielding more accurate, objective, and meaningful performance evaluation results.

To validate the superiority of the proposed model in short-term bus load forecasting, multiple representative baseline models were selected for performance evaluation, including ARIMA, BiLSTM, and TCN, as well as their variants combined with attention mechanisms (Att) and convolutional modules. These models cover different paradigms of deep learning and traditional statistical modeling, providing a comprehensive reflection of the current mainstream methods’ performance differences in short-term load forecasting tasks.

All models employ a multi-step forecasting approach, directly outputting predictions for the next 96 time points. To visually compare the predictive capabilities of each model in practical scenarios, a single day’s data was randomly selected from the test set, and the predicted values were compared against the true values, as shown in Figures 9–13. The graphical results clearly demonstrate the relative strengths and weaknesses of each model’s forecasting accuracy across different user load types, offering an intuitive basis for subsequent analysis.

Compared with widely used baseline models such as ARIMA, BiLSTM, and TCN, the experimental results demonstrate that although all models exhibit reasonable predictive capabilities for short-term load forecasting, the proposed Attention-enhanced Inception-BiGRU-Seq2Seq (AIBS) model consistently outperforms them across all evaluation metrics.

This superior performance stems from several key architectural innovations. First, the adoption of a Seq2Seq forecasting framework enables the model to capture long-range temporal dependencies, facilitating more accurate mapping between input and output sequences. Second, the integration of an attention mechanism dynamically emphasizes the most informative time steps, mitigating the information dilution problem often encountered in long-sequence forecasting. In parallel, the Inception module enriches feature representation through muti-scale convolution kernels, allowing the model to capture both fine-grained local variations and broader contextual patterns in the input data.

In addition, adaptive hyperparameter tuning with the GWO automates the search for critical parameters such as learning rate and hidden layer size, thereby improving both convergence speed and generalization capability. Consequently, the AIBS model produces predicted load curves that closely align with actual measurements, maintaining high accuracy even during periods of sharp load fluctuations, such as those induced by extreme weather conditions.

These findings underscore the robustness and adaptability of the proposed AIBS approach and highlight its practical value for real-world power system operation and decision-making under increasingly complex and volatile load patterns.

Figures 14–18 present the absolute error (AE) curves of the proposed AIBS model for the 96-step multi-horizon forecasting task. In these figures, the horizontal axis represents forecast time steps, while the vertical axis denotes the absolute error between predicted and actual load values. As shown, the error fluctuations remain relatively small across most forecast horizons, exhibiting a smooth and stable trend. Only a few time steps show slightly larger error spikes, likely caused by sudden load changes or highly volatile operating conditions.

These results indicate that the AIBS model delivers stable fitting performance, with prediction deviations generally maintained within an acceptable error range.

Although minor error fluctuations occur at certain points—primarily due to abrupt load shifts or extreme weather-induced demand surges—the overall results confirm the robustness and resilience of the proposed model when addressing the complex, nonlinear, and nonstationary characteristics of short-term load data.

Figures 19–23 display scatter plots depicting the multi-day short-term load forecasts generated by the proposed model. In these plots, the horizontal axis corresponds to actual load values, while the vertical axis denotes the forecasted values. These visualizations are designed to illustrate the linear correlation between the model’s predictions and the real-world observations. Notably, the data points cluster closely around the fitted line—this dense aggregation signals a high level of consistency between predicted and actual load values, thereby highlighting the model’s robust fitting performance.

Such a concentrated point distribution further validates the model’s efficacy in capturing both the overall trends and short-term fluctuations inherent in load data. For quantitative assessment, the fitting evaluation metrics (specifically R values) for different load categories are 0.97338, 0.98091, 0.97368, 0.98232, and 0.98546, all of which are very close to 1. These results confirm that the proposed model delivers exceptionally high prediction accuracy and stability across diverse forecasting scenarios. As a key indicator, a higher R value reflects a stronger linear correlation between predicted and actual values, which in turn underscores the model’s capacity to accurately replicate real-world load variations.

Table 5 summarizes the prediction errors of various models across multiple load forecasting scenarios and evaluation metrics. The results clearly demonstrate that the proposed AIBS model consistently attains the lowest prediction errors in all evaluated cases, significantly outperforming traditional methods such as ARIMA and deep learning benchmarks including BiLSTM and TCN.

A key strength of the AIBS model lies in its superior adaptability to complex and highly volatile load patterns, particularly evident in industrial and commercial settings—such as real estate companies, sand and gravel plants, and shopping malls—where load dynamics are strongly affected by external weather, operational schedules, and abrupt demand surges. In these scenarios, the AIBS model achieves substantial error reductions, effectively capturing nonlinear and dynamic load behaviors beyond the capabilities of competing approaches.

For example, in the Shopping Mall scenario, ARIMA yields an RMSE of 61.41 and MAE of 45.00, underscoring the limitations of traditional linear models in handling nonlinear, fluctuating loads, especially under extreme conditions like heatwaves. Progressive improvements are observed moving from BiLSTM and TCN, to their attention-augmented variants, and further to CNN-integrated models. Nevertheless, even the best-performing baseline (BiLSTM + ATT + CNN) exhibits markedly higher errors compared to AIBS, highlighting the latter’s breakthrough in accuracy via its innovative design and optimization.

A particularly notable case is the Tech Company scenario, characterized by high-temperature and high-heat operation. Here, AIBS achieves an RMSE of 0.1697 and MAE of 0.3186, representing over 70% error reduction relative to ARIMA (RMSE = 0.6745, MAE = 0.7829). This underscores the model’s robustness and precision under extreme climatic conditions, providing strong technical support for real-time load dispatch, emergency management, and demand-side strategies during heatwave events.

In summary, the proposed AIBS model exhibits clear superiority over both traditional statistical and advanced neural network methods in short-term load forecasting. Its predictions closely align with actual load trajectories and accurately reflect temporal fluctuations across diverse scenarios. Conversely, other benchmark models, despite some predictive capability, suffer from bias and structural distortions under complex load conditions, resulting in comparatively inferior performance. These findings validate the effectiveness, robustness, and practical value of AIBS as a reliable and generalizable forecasting solution for modern power systems facing increasingly volatile operating environments.

To comprehensively validate the effectiveness and contribution of each key component within the proposed Attention-enhanced Inception-BiGRU-Seq2Seq (AIBS) model, a series of systematic ablation experiments were conducted. These experiments spanned multiple forecasting horizons—including 4, 8, 16, 48, 96, and 192 steps—to assess predictive accuracy across diverse user load forecasting tasks while progressively removing or substituting critical modules. This approach enabled a detailed examination of the individual and combined impacts of components such as the Inception module, BiGRU layers, improved DSW attention mechanism, and hyperparameter optimization via the GWO on overall model performance.

Beyond accuracy evaluation, the experiments also analyzed computational efficiency, particularly comparing training and inference times for the 96-step prediction task with and without the improved DSW attention. Quantitative accuracy results for the various model variants across forecasting horizons are presented in Figures 24–28, with training and inference time comparisons summarized in Table 6. Supplementary Table SA1 in the appendix shows the impact of different modules on prediction accuracy.

The results reveal a clear trend: prediction errors increase as the forecasting horizon extends across all user load categories, reflecting the inherent difficulty of long-sequence time series forecasting. Nevertheless, the proposed AIBS model consistently outperforms its ablated counterparts and alternative architectures at all horizons, demonstrating robust scalability and accuracy. Notably, it shows pronounced improvements in ultra-short-term forecasting (4-step ahead), crucial for real-time operational decisions, while maintaining superior performance in short-to mid-term multi-step forecasting scenarios (8, 16, 48, 96, and 192 steps), confirming its adaptability to diverse temporal requirements.

From a computational standpoint, integrating the improved DSW attention mechanism yields significant efficiency gains, reducing total training time by approximately 7,678 s and inference time by around 5 s compared to conventional attention mechanisms with similar functions. This highlights the DSW attention’s effectiveness in expanding the receptive field while controlling computational costs, thereby enhancing the model’s suitability for deployment in resource-constrained or real-time forecasting contexts.