Short-term load forecasting for industrial users is often limited by scarce local data and heterogeneous load patterns, while cross-factory collaboration raises serious privacy concerns due to production-sensitive information. This paper proposes a privacy-preserving federated forecasting framework for industrial short-term load prediction that integrates meta-learning and differentiated differential privacy allocation. The framework is built on federated learning, where factories perform local training and upload encrypted model updates for secure aggregation, ensuring data availability without data visibility. To address negative transfer and poor personalization caused by load heterogeneity, a meta-learning strategy is introduced to enable rapid adaptation to new factories or operating conditions using only a small amount of local data. In addition, a differentiated privacy-weight mechanism is designed to dynamically allocate privacy budgets based on data contribution and sensitivity, achieving a better trade-off between privacy protection and forecasting accuracy. Experiments on real-world industrial load datasets show that the proposed method reduces MAPE by 1.99 percentage points compared with independent training, and by 2.63 percentage points in new operating-condition scenarios, demonstrating its effectiveness for secure and collaborative industrial load forecasting.
federated learningindustrial load forecastingmeta-learningprivacy preservingshort-term load forecastingThe author(s) declared that financial support was not received for this work and/or its publication.section-at-acceptanceEnergy EfficiencyIntroduction
Industrial load forecasting is a fundamental basis for the secure and stable operation and refined management of power systems, playing a critical role in power grid dispatching, demand response, energy efficiency assessment, and renewable energy integration (Duan et al., 2026; Hasan et al., 2025). Compared with residential and commercial users, industrial electricity consumption is highly coupled with production schedules, process flows, and equipment operating conditions, resulting in load variations that are often abrupt and highly stochastic, thereby imposing more stringent requirements on short-term load forecasting (STLF) (Liu et al., 2023). However, due to frequent switching of production conditions and the diversity and complexity of industrial processes, industrial loads exhibit pronounced non-stationary and strongly nonlinear characteristics, making it difficult for traditional statistical models and single data-driven approaches to accurately capture their dynamic evolution patterns (Hu and Man, 2023; He et al., 2025). Moreover, constrained by the operating cycles of individual factories and the level of data accumulation, the historical load data available for modeling are often limited in scale, resulting in insufficient model training and restricted generalization capability, with prediction performance deteriorating significantly under new operating conditions or abnormal scenarios. How to fully exploit common patterns across multiple entities and scenarios while ensuring industrial data security has thus become a key issue in improving the accuracy of industrial short-term load forecasting. Cross-factory collaborative modeling, by integrating multi-source industrial load features, is expected to alleviate the performance bottlenecks caused by data scarcity and to provide new research perspectives for building more robust and high-accuracy industrial load forecasting models.
For industrial short-term load forecasting tasks, existing methods can generally be categorized into two types according to the number of participating entities: independent forecasting (Li et al., 2025) and cross-user centralized collaborative forecasting (Huang et al., 2025). Independent forecasting is typically based on data from a single factory and employs statistical models such as ARIMA and regression models, or machine learning and deep learning models such as LSTM and Transformer (Biswal et al., 2024; Wazirali et al., 2023). Walther et al. (2019) proposed a GBRT-based short-term load forecasting method that integrates feature selection, feature engineering, and hyperparameter tuning, achieving high-accuracy 15-min industrial load forecasting and effectively addressing the difficulties of model construction and optimization under high-dimensional multi-source features. Zhu et al. (2023) developed an industrial load forecasting approach that combines load change rate features with a firefly-algorithm-optimized extreme learning machine and AdaBoost ensemble learning, enabling effective characterization of the nonlinear properties of industrial loads and significantly reducing forecasting errors while overcoming the limitations of traditional models in feature representation and parameter optimization. Zhang et al. (2025) addressed the strong nonlinearity and severe fluctuations of industrial building loads by proposing a short-term load forecasting method that integrates multiple signal decomposition techniques with deep hybrid networks; by combining VMD and CEEMDAN decomposition with sample entropy–based reconstruction, the method effectively reduces noise interference and extracts multi-scale load features. Walser and Sauer (2021) proposed a hybrid machine learning model for factory-level load forecasting, which combines typical load profile analysis with convolutional neural networks and leverages near-real-time high-resolution multidimensional data to achieve robust and high-accuracy short-term load forecasting, effectively overcoming the limitations of traditional models in terms of real-time performance, multi-source sub-meter data fusion, and cross-factory generalization. Majeske et al. (2025) introduced an industrial energy forecasting framework based on dynamic knowledge graphs and attention-based RNNs, achieving high-accuracy and interpretable energy consumption prediction, significantly outperforming conventional RNN models while maintaining suitability for edge deployment. However, statistical models are constrained by assumptions of linearity and stationarity, and although machine learning and deep learning models can capture nonlinear relationships, they rely heavily on sufficient high-quality historical data and are prone to overfitting with limited generalization capability in scenarios characterized by frequent industrial load pattern switching, high noise, and data scarcity. Overall, independent forecasting is easy to implement and deploy because it only requires single-factory data and avoids cross-organization data sharing. It can be tailored to a specific factory’s operational schedule and equipment characteristics, thus achieving high accuracy when sufficient high-quality historical data are available. However, its performance is often constrained by data scarcity and non-stationary load behaviors in industrial scenarios. In particular, frequent production-mode switching, strong noise, and abrupt changes may lead to limited generalization, high sensitivity to feature drift, and overfitting, especially for complex deep models trained on small datasets.
To overcome the limitations of single-site data, an alternative line of research explores collaborative learning across multiple factories to exploit shared patterns and enlarge the training set. To alleviate the performance bottlenecks caused by data scarcity, cross-user centralized collaborative forecasting trains a unified model by aggregating load data from multiple factories, thereby expanding the sample size, extracting common patterns, and improving forecasting accuracy and robustness (Manzoor et al., 2024; Fan et al., 2024). Liu et al. (2025) proposed a self-allocated Kolmogorov–Arnold Network (KAN) method for multi-energy load forecasting in integrated energy systems, which employs multi-decoder Informer encoding combined with random sample self-composition and correlation feature self-allocation modules to achieve multivariate fusion and co-decoding, thus enhancing predictive performance under coupled and uncertain scenarios. Liao et al. (2024) developed a bi-level multi-task learning (BiMTL) approach for multi-energy load forecasting in integrated energy systems, which enhances local feature representation through an improved wavelet packet decomposition and performs reconstruction and error correction of prediction results in the second-level shared learning stage. Experimental results demonstrate that the constructed BiMTL-LSTM model significantly outperforms existing methods. Cross-user centralized collaborative forecasting can significantly improve robustness by aggregating multi-factory data, enabling the model to learn common load dynamics and reducing variance under data scarcity. Nevertheless, it faces practical challenges: (i) centralized data collection may violate privacy constraints and data governance policies; (ii) industrial loads are highly heterogeneous across factories, so a unified model may suffer from negative transfer or biased learning toward dominant participants; and (iii) communication, storage, and continuous updating of large-scale multi-factory datasets increase engineering complexity and cost. These limitations motivate the need for collaborative forecasting paradigms that can leverage cross-user knowledge while addressing privacy and heterogeneity.
To address the privacy and compliance challenges inherent in centralized collaborative forecasting, research efforts have gradually shifted toward collaborative modeling under privacy-preserving conditions (Kaur et al., 2025; Shen et al., 2024). Achieving cross-entity information collaboration without directly sharing raw load data has thus become a key research direction. Privacy-preserving collaborative learning frameworks represented by federated learning effectively reduce the risk of industrial data leakage through local training and model parameter exchange, providing a feasible pathway for multi-factory joint modeling. Khalil et al. (Li et al., 2024) proposed a privacy-preserving federated learning approach for building energy load forecasting by incorporating differential privacy (DP) and a POWER-SELECTION protocol, which significantly reduces communication overhead while safeguarding user electricity consumption privacy; experimental results show that the method achieves an effective balance between improved forecasting accuracy, bandwidth efficiency, and provable privacy protection. Wang and Li (2022) introduced a cloud–edge collaborative short-term load forecasting method for smart grids, which constructs a pool of pre-trained models in the cloud and performs lightweight retraining and optimal model selection at the edge, thereby improving forecasting accuracy while reducing edge-side computational burden; the effectiveness of the method was validated through extensive experiments, while also addressing data privacy and personalization through pre-training transfer and hierarchical edge training. Chen et al. (2024) proposed an improved Transformer model based on federated learning for estimating behind-the-meter photovoltaic power generation at the community level without centralized sharing of raw data, effectively mitigating privacy risks associated with centralized training; by combining locally retained parameters with exchanged parameters in a dual-layer structure, the method enables secure collaborative modeling of common features across multiple nodes while protecting user-side data privacy. Huang et al. (2024) developed a federated learning–based load forecasting method for energy aggregation service providers, which achieves accurate prediction through weighted collaborative training and parameter sharing among energy entities without exchanging raw data. Nevertheless, due to the pronounced differences in factory operating modes and the high heterogeneity of industrial load characteristics (Yu et al., 2016), existing methods still suffer from limitations in model generalization capability, adaptability to new operating conditions, and the trade-off between privacy protection strength and forecasting performance, indicating the need for further research and improvement.
Compared with existing studies, the proposed method provides a unified solution to three key challenges in industrial STLF, namely, data scarcity, cross-factory heterogeneity, and privacy–accuracy trade-off. Unlike independent forecasting that is often constrained by limited local data and weak generalization under operating-condition shifts, our framework enables cross-factory knowledge sharing while preserving personalization. In contrast to centralized collaborative forecasting that requires collecting raw industrial data and may violate privacy and compliance requirements, the proposed approach follows a federated learning paradigm so that raw load data remain on-site and only protected updates are exchanged. Moreover, compared with conventional federated forecasting that mainly relies on parameter averaging and is prone to negative transfer under heterogeneous factories, we incorporate meta-learning for rapid adaptation and an adaptive DP allocation mechanism to balance privacy protection and forecasting performance. The main contributions of this paper are summarized as follows:
To mitigate negative transfer and insufficient personalization caused by heterogeneous industrial load patterns, a meta-learning algorithm for rapid cross-factory adaptation is proposed, enabling the global model to quickly adapt to new operating conditions or new factories with only a small amount of local data, thereby enhancing personalized forecasting performance and generalization capability.
To alleviate the conflict between privacy protection strength and model performance, an improved federated learning algorithm incorporating DP weights is designed. By jointly quantifying the data contribution and privacy sensitivity of each factory, the proposed method dynamically allocates DP budgets and performs weighted aggregation, achieving an adaptive trade-off between privacy protection and forecasting accuracy.
Experiments on real-world industrial load datasets show that the proposed method reduces MAPE by 1.99 percentage points compared with independent training, and by 2.63 percentage points in new operating-condition scenarios, demonstrating its effectiveness for secure and collaborative industrial load forecasting.
MethodologiesOverall framework of the model
As illustrated in Figure 1, the proposed privacy-preserving cross-scenario collaborative industrial short-term load forecasting framework contains three tightly coupled components under the federated learning paradigm: (i) a local meta-learning module deployed at each factory (client), (ii) an adaptive privacy allocation module executed on the client side to protect the shared updates, and (iii) a federated aggregation module on the server side. These modules operate collaboratively under the federated learning paradigm to enable the sharing and transfer of load forecasting capabilities across factories while ensuring data privacy and security.
Local Meta-learning module. On the local side, each industrial factory acts as an independent client and performs meta-learning training based on its own historical load data Di. Specifically, to improve generalization across different operating conditions within the same factory, Di is organized into a set of scenario-driven learning tasks. For each task, the data are split into a support set and a query set: the support set is used for inner-loop adaptation, while the query set is used for outer-loop evaluation to compute the meta-objective and obtain a task-level meta-gradient. After completing task-level training over multiple tasks, the client produces a single update that summarizes how the meta-model should be adjusted to adapt quickly across local scenarios. This update is the meta-gradient Δθi, which summarizes high-information-density transferable knowledge about how the forecasting model should adapt across scenarios. Although this procedure enables effective feature extraction from limited local data, gradient-level sharing may still introduce potential privacy leakage risks; therefore, raw time-series data never leave the factory, and only meta-gradient information is prepared for communication.
Adaptive Privacy Allocation module. Subsequently, the locally generated meta-gradient Φi is fed into the adaptive privacy allocation module for privacy-preserving processing before transmission. This module quantitatively evaluates two key factors: (i) the data contribution (utility) of the client update to global modeling and (ii) the corresponding privacy sensitivity (risk) associated with sharing gradient information. Based on these two assessments, an adaptive “privacy–performance” regulation mechanism assigns a client-specific DP budget. The meta-gradient is then clipped and perturbed with calibrated noise to obtain a privacy-protected update. Intuitively, factories with higher privacy sensitivity are allocated stricter privacy budgets, whereas updates with greater contribution are allowed to preserve more effective features under controllable privacy constraints, thereby enabling the extraction of shareable information. The protected update is further encrypted for secure transmission to the server.
Federated Aggregation module. At the server side, the federated aggregation module receives encrypted updates from all clients and performs secure aggregation to prevent the exposure of individual client updates. The server then conducts privacy-aware weighted fusion to iteratively update the global meta-model parameters. By integrating heterogeneous client updates in a differentiated manner, the server balances forecasting performance improvement and privacy protection. The updated global model is subsequently redistributed to all factories for the next round of local meta-learning, forming a closed-loop collaborative optimization process of “local adaptation–privacy control–global aggregation.”
Overall structure of the model.
Diagram illustrating a federated meta-learning process with three sections: local meta-learning using multi-factory client data for model updates; adaptive privacy allocation balancing privacy and performance through evaluation; federated aggregation module updating encrypted meta gradients for secure sharing.
Overall, the proposed framework achieves “data remaining on-site with collaborative model learning” at the architectural level, enhances the model’s rapid cross-scenario adaptation capability through meta-learning, and alleviates the conflict between privacy protection and forecasting performance via DP allocation, thereby providing a systematic solution for secure and efficient collaborative modeling in industrial load forecasting scenarios.
Meta-learning model based on autoformer
To combine Autoformer with meta-learning, this paper uses Autoformer as the backbone forecasting network fθ in the meta-learning framework, and treats each factory as an independent task. Specifically, the series decomposition and auto-correlation mechanism of Autoformer are retained to extract trend–seasonal representations and capture long-term periodic dependencies, while the model parameters θ of the entire Autoformer are optimized in a bi-level meta-learning manner. In each meta-training round, Autoformer first performs a few inner-loop updates on the support set of a factory to obtain task-adapted parameters θi′, and then computes the meta-gradient using the query set to evaluate post-adaptation performance. The server (or meta-learner) aggregates meta-gradients across factories to update the shared initialization θ, so that the learned Autoformer initialization can rapidly adapt to unseen factories or new operating conditions with only limited local data and a small number of gradient steps.
Autoformer
To effectively capture the significant trend and cyclical characteristics in short-term industrial load sequences and enhance the model’s adaptability in small-sample and new operating condition scenarios, this paper selects Autoformer as the base model in the meta-learning framework. By employing series decomposition and an Auto-Correlation Mechanism, Autoformer achieves efficient modeling of long-term dependencies (Ouyang et al., 2025; Ban et al., 2024), demonstrating stronger stability and generalization capabilities compared to traditional Transformer models in time series forecasting tasks. The structure of the Autoformer model is shown in Figure 2.
Series Decomposition Mechanism: Industrial load time series typically contain both long-term variation trends and cyclical fluctuations induced by production rhythms, shift schedules, and other factors, exhibiting pronounced non-stationary characteristics. Directly modeling the raw series requires the model to simultaneously learn patterns across different temporal scales, which not only increases modeling difficulty but also tends to lead to unstable training. To address this, Autoformer introduces a series decomposition mechanism during the encoding stage, explicitly separating the original load series into trend and seasonal components, thereby achieving decoupled modeling of the complex temporal structure. Specifically, the trend component captures the slowly evolving long-term characteristics of the load over time, such as adjustments in production scale or changes in equipment operating states; while the seasonal component reflects short-term fluctuation patterns caused by periodic production behaviors. By employing moving average operations to extract the trend component, high-frequency noise and short-term disturbances can be effectively filtered out, allowing the remaining seasonal component to concentrate more on expressing the cyclical structure. This decomposition process not only reduces the non-stationarity of the series but also enables the model to separately model long-term trends and short-term dynamics at different levels, thereby enhancing the stability and generalization capability of predictions. Given an industrial load time series X=x1,x2,...,xT, Autoformer first decomposes it into trend and seasonal components, as shown in Equation 1. The trend component Xtrend is obtained through a moving average operation, as presented in Equation 2. The seasonal component Xseason is calculated according to Equation 3. This decomposition approach helps reduce non-stationarity and enables the model to focus more on the cyclical structure and dynamic variation patterns.
Auto-Correlation Attention Mechanism: In industrial load forecasting tasks, current load values often exhibit significant correlations with loads at certain historical moments, and such correlations typically present periodic repetition characteristics, such as daily or weekly cycles. The self-attention mechanism of the traditional Transformer models these relationships through pairwise similarities among all time points, resulting in high computational complexity and difficulty in explicitly capturing periodic dependencies. To address this issue, Autoformer adopts an auto-correlation attention mechanism centered on delay correlations, which directly models the periodic structures in time series. This mechanism first computes the correlation degree of the seasonal component under different time delays, measuring the similarity of the sequence across varying time offsets. By selecting several delay terms with the highest correlation, the model can automatically identify the most significant cycle lengths in the sequence and incorporate corresponding historical information into the representation of the current moment through weighted aggregation. This delay-based aggregation approach avoids pairwise computations across all time points, not only reducing computational complexity but also better aligning with the inherent “periodic repetition” pattern of time series. During the encoding stage, Autoformer replaces the conventional self-attention with the auto-correlation mechanism, enabling explicit modeling of periodic dependencies in time series. Its core idea is to leverage delay correlations for capturing long-term dependencies in the sequence, as shown in Equation 4. Features are aggregated via weighted summation using the K time delays τk with the highest correlations, as shown in Equation 5.
X=Xtrend+XseasonXttrend=1k∑j=t−⌊k/2⌋t+⌊k/2⌋xjXtseason=xt−XttrendRτ=∑t=1T−τXtseason·Xt+τseasonAutoCorrX=∑k=1Kαk·RollX,τkwhere, X denotes the input industrial load time series with length T, and xt is the load value at time index t. Xtrend and Xseason denote the trend and seasonal components, respectively. k denotes the moving-average window size, and ⌊·⌋ is the floor operator. τ denotes the time delay, and Rτ denotes the delay correlation score computed on the seasonal component. K denotes the number of selected delays, and τk denotes the k-th delay among the top-K delays with the highest correlation. αk denotes the normalized correlation weight corresponding to τk, and Roll· represents the sequence shift operation along the time dimension by τk steps. AutoCorrX denotes the aggregated representation obtained via the weighted summation of the shifted sequences.
The structure of autoformer.
Block diagram illustrating the Autoformer model for time series forecasting, showing encoder and decoder blocks with auto-correlation, series decomposition, feed forward layers, and initialization of seasonal and trend-cyclical components leading to prediction.Meta learning model
In cross-factory industrial short-term load forecasting, the load distributions of different factories exhibit significant heterogeneity due to factors such as production scale, process flow, and operational regimes. Directly applying a unified global model often struggles to simultaneously account for the distinct characteristics of each factory, and may even lead to negative transfer. To address this, this paper introduces a meta-learning mechanism, enabling the model to learn shared priors across multiple factory tasks and achieve rapid adaptation in scenarios with new operating conditions or new factories. The core of meta-learning lies in learning a well-transferable parameter initialization, allowing the model to quickly adapt to new tasks with only a small amount of local data and a few update steps (Yuan et al., 2026; Xiao et al., 2022).
Within the meta-learning framework, each factory is defined as a task Ti, with its local dataset Di divided into a support set Disup and a query set Diqry. The model parameters θ no longer aim solely for optimal fitting on all data but serve as an initialization point with good adaptability to task variations. Specifically, the model first undergoes rapid intra-task updates on the support set, as shown in Equation 6. This process simulates the model’s quick adaptation behavior after receiving a small amount of new data. Subsequently, the updated model parameters θi′ are evaluated on the query set, and the meta-gradient relative to the initial parameters θ is computed, as shown in Equation 7. This meta-gradient captures the sensitivity of the model’s generalization performance to changes in the initial parameters after intra-task updates, representing key information shared across tasks. After repeating the above process across multiple factory tasks, the global parameters are updated by aggregating the meta-gradients from each task, as shown in Equation 8. Through this update strategy, the model progressively learns a parameter initialization that enables it to achieve superior predictive performance with only a few gradient updates when faced with different factories or new operating conditions. Combining Autoformer’s strengths in long-term dependency modeling and series decomposition, the meta-learning mechanism further enhances the model’s generalization capability and adaptation efficiency in small-sample and highly heterogeneous industrial scenarios.θi′=θ−α∇θLTifθ,DisupΦi=∇θLTifθi′,Diqryθ←θ−β∑i∈SωiΦiwhere Ti denotes the i-th factory task, Disup and Diqry denote the corresponding support set and query set, respectively. θ denotes the global meta-parameters (initialization) before adaptation, fθ is the forecasting model parameterized by θ, θi′ denotes the task-adapted parameters after the inner-loop update on Disup, LTi· represents the task-specific prediction loss function, ∇θ denotes the gradient operator with respect to θ, Φi denotes the meta-gradient computed from the query loss after adaptation, α denotes the intra-task learning rate, S denotes the set (mini-batch) of sampled tasks in each meta-update, β denotes the meta-learning rate, and wi denotes the task weight coefficient for aggregating the meta-gradients from different tasks.
Federated learning model integrating differential privacy algorithm
During the federated collaborative forecasting process, although the original industrial load data remain locally stored, model gradients or parameter updates may still risk leaking sensitive production information of the factories. To further enhance the privacy and security of the system, this paper introduces a DP mechanism into the federated learning framework (Sarantinopoulos et al., 2025) and designs an adaptive privacy allocation strategy, aiming to ensure data security while minimizing the impact of privacy noise on model performance. The model structure is illustrated in Figure 3. Figure 3 depicts a standard client–server federated learning architecture augmented with client-side DP and server-side secure aggregation. Each factory trains the local forecasting model using its own on-site industrial load data. Before any update leaves the factory, the DP module processes the locally computed meta-gradient/parameter update: (i) it bounds the sensitivity of the update by clipping, ensuring that a single sample (or a small subset of samples) cannot dominate the update; (ii) it injects calibrated random noise into the clipped update, so that the shared information becomes statistically indistinguishable with respect to the presence/absence of any individual record, thereby reducing the risk of inferring sensitive production characteristics from communicated gradients. The resulting DP-protected update is then encrypted and transmitted to the aggregation server (Step 1), meaning that the server never receives raw data or unprotected gradients. At the server side, the aggregation server performs secure aggregation (Step 3), which ensures that only the aggregated update over multiple factories can be recovered, while individual client updates remain hidden. The server then computes the global model update based on the aggregated (DP-protected) information and sends the updated global parameters back to each factory (Step 2). Finally, each factory updates its local model with the received global parameters and continues the next local training round (Step 4). Therefore, the DP algorithm plays a critical role by providing a privacy-preserving “filter” on the client side—transforming potentially sensitive gradients into protected shareable signals—while secure aggregation prevents exposure of single-client updates at the server side. Together, they enable collaborative model training with “data staying on-site” and privacy-aware information exchange.
Structure of federated learning model integrating differential privacy algorithm.
Diagram showing a federated learning process where multiple factories each apply differential privacy to local neural networks, send encrypted gradients to a central aggregation server for secure aggregation, and receive model updates, with steps labeled for encrypted gradients, model updates, secure aggregation, and updating models.
DP protects data by injecting random noise during the model update process, making it difficult for an attacker to determine whether a specific sample participated in the training (Wang and Kang, 2026). For any adjacent datasets D and D′, if a randomized mechanism M· satisfies Equation 9, then the mechanism is said to satisfy ε-DP. In this paper, the DP mechanism is applied to the meta-gradients Φi uploaded by each factory to prevent the inference of local load data characteristics from gradients. To limit the impact of individual factory model updates on privacy leakage risk, the meta-gradients are first clipped by their L2 norm, as shown in Equation 10. Subsequently, Gaussian noise is injected into the clipped gradients, as shown in Equation 11.
There are significant differences in data volume, data quality, and privacy sensitivity among different factories in federated learning. To avoid the issues of “over-protection” or “privacy deficiency” caused by a unified privacy budget, this paper proposes an adaptive privacy allocation strategy, assigning differentiated privacy budgets to each factory. First, a data contribution metric for factory i is defined, as shown in Equation 12. This metric reflects the importance of the factory’s updates to the optimization of the global model. Simultaneously, a privacy sensitivity coefficient ρi is introduced to characterize the factory’s demand intensity for data privacy protection. Considering both contribution and privacy sensitivity, the privacy budget for factory i is allocated as shown in Equation 13. This design allows factories with higher contributions and lower privacy sensitivity to receive relatively relaxed privacy constraints, thereby reducing noise interference, while factories with higher privacy sensitivity automatically obtain stronger protection. Finally, on the server side, the differentially private meta-gradients are weighted and aggregated to update the global model parameters, as shown in Equation 14, where the update weight wi is related to the privacy budget, as shown in Equation 15. This mechanism further mitigates the adverse impact of high-noise updates on the global model, achieving a dynamic balance between privacy protection and predictive performance.PrMD∈S≤eεPrMD′∈SΦ¯i=Φimax1,Φi2CΦ∼i=Φ¯i+N0,σi2C2Iηi=Φi2∑j∈SΦj2εi=εtotal·ηiρiθ←θ−β∑i∈SωiΦ∼iωi=εi∑j∈Sωjwhere ε is the privacy budget, with smaller values indicating higher privacy protection strength. C is the clipping threshold, and σi denotes the noise intensity, which is inversely proportional to the privacy budget εi of factory i. I represent the identity matrix, whose dimensionality matches that of the meta-gradient vector Φ¯i.
Case studyData description and model configuration
This study considers six industrial users connected to a regional power grid in northern China, including five conventional industrial users and one newly connected industrial user. The newly connected user has a limited amount of historical load data and is mainly used to evaluate the rapid adaptation capability of the proposed federated meta-learning model under small-sample and new operating-condition scenarios. The time span of the load data for the industrial users ranges from 1.5 to 3 years, with installed capacities between 128 MW and 381 MW, as detailed in Table 1. In this study, the privacy protection level (High/Medium/Low) reflects the sensitivity of each factory’s load data with respect to production confidentiality, and is implemented by allocating differentiated differential-privacy budgets via privacy-weight coefficients (High/Medium/Low: 0.5/0.3/0.2) under a fixed gradient clipping threshold of 1.0 and a noise multiplier of 0.6.
Overview of industrial user data.
User ID
User type
Data time span
Installed capacity (MW)
Privacy level
F1
Conventional user
3 years
310
High
F2
Conventional user
2.5 years
128
Medium
F3
Conventional user
2 years
258
Medium
F4
Conventional user
2 years
219
Low
F5
Conventional user
1.5 years
381
Low
F6
Newly connected user
1 month
292
High
To capture the temporal evolution characteristics of industrial loads, a short-term load forecasting task is constructed. The forecasting resolution is 15 min, and the prediction target is the active power load at the next time step. The model input features include historical load sequences, meteorological variables (e.g., ambient temperature and humidity), and timestamp features (such as time of day and day of week), which enhance the model’s ability to learn periodic patterns and respond to external disturbances.
Considering the heterogeneity among industrial users in terms of data sensitivity and privacy requirements, users are categorized into three privacy protection levels—high, medium, and low—according to the confidentiality of their production-related load data. Correspondingly, differentiated privacy-preserving strategies are adopted in the federated learning process. For each user, the dataset is split into training, validation, and test sets with a ratio of 70%/15%/15%. During data preprocessing, outliers are removed, missing values are imputed, and all input features are normalized using Min–Max scaling to ensure a consistent numerical range.
Model training is conducted in a unified computational environment, with an NVIDIA GPU as the hardware platform and Python-based deep learning frameworks as the software environment. The forecasting model adopts Autoformer as the base time-series prediction network, upon which a collaborative framework integrating federated learning and meta-learning is constructed. The hyperparameters of Autoformer, as well as those related to federated learning and meta-learning, follow default configurations that have been validated in classical literature to ensure fairness and reproducibility of model comparisons. The detailed parameter settings are summarized in Table 2.
Model architecture and training parameters.
Module
Parameter
Value
Autoformer
Input sequence length
96 (24 h with 15-min resolution)
Forecast horizon
96 (24 h with 15-min resolution)
Encoder layers
2
Decoder layers
1
Model dimension
512
Number of heads
8
Feed-forward dimension
2048
Dropout
0.1
Activation function
GELU
Training
Optimizer
Adam
Initial learning rate
0.0001
Batch size
32
Epochs
20
Learning rate scheduler
Cosine annealing
Federated learning
Number of clients
6
Aggregation algorithm
FedAvg
Communication rounds
50
Local update epochs
1 (per round)
Client participation ratio
1.0 (full participation)
Meta-learning
Meta-learning algorithm
Reptile
Inner-loop steps
5
Inner-loop learning rate
0.001
Outer-loop learning rate
0.0001
Differential privacy
Gradient clipping threshold
1.0
Noise multiplier
0.6
Privacy budget weights
High/Medium/Low: 0.5/0.3/0.2
Evaluation metrics
To enable a fair and comparable evaluation of model performance in privacy-preserving cross-factory collaborative forecasting scenarios, three error metrics are adopted: mean absolute percentage error (MAPE), normalized root mean square error (NRMSE), and normalized mean absolute error (NMAE). Given the heterogeneity among factories in terms of load scale, production schedules, and data distributions, normalized and dimensionless metrics are employed to mitigate the dominant influence of capacity differences on evaluation results. Specifically, all error metrics are normalized by the installed capacity of each industrial user to ensure consistency and comparability across factories with different load scales. This allows a more objective assessment of the prediction accuracy improvements achieved by the federated meta-learning framework under the constraint of “data available but not visible”.
In practical evaluation, each factory computes the performance metrics locally on its own test set, and only the scalar metric values are uploaded for statistical aggregation. This design avoids the disclosure of raw load sequences or fine-grained information that could potentially be reverse-engineered, thereby ensuring privacy preservation throughout the evaluation process.
Let N denote the number of samples in the test set, yt the ground-truth load at time t, y^t the corresponding predicted load, and yc the installed capacity of the user. The three metrics are defined as Equations 16–18.MAPE=100N∑t=1Nyt−y^tytNRMSE=1N∑t=1Nyt−y^t2ycNMAE=1N∑t=1Nyt−y^tyc
Multi-user forecasting results and accuracy analysis
Tables 3–5 report the comparative results of the proposed federated learning + meta-learning collaborative framework and traditional independent forecasting on five industrial users (F1–F5) under three error metrics: MAPE, NRMSE, and NMAE. As observed, the proposed framework consistently outperforms independent training across all three backbone models, with stable and significant accuracy gains.
MAPE comparison between the proposed method and independent forecasting (%).
Method
Model
F1
F2
F3
F4
F5
FL + ML
Autoformer
3.61
3.74
3.69
3.93
3.85
Transformer
3.78
3.89
3.84
4.03
3.89
TimeMixer
3.98
4.19
3.98
4.08
3.88
Independent
Autoformer
5.78
5.66
5.80
5.56
5.97
Transformer
4.87
5.54
5.21
5.34
5.19
TimeMixer
6.07
6.01
6.13
6.09
5.82
NRMSE comparison between the proposed method and independent forecasting (%).
Method
Model
F1
F2
F3
F4
F5
FL + ML
Autoformer
8.50
11.70
8.36
14.63
17.66
Transformer
9.01
12.42
8.52
14.88
16.18
TimeMixer
9.05
13.57
9.21
14.37
15.80
Independent
Autoformer
12.68
16.74
13.36
18.61
22.65
Transformer
10.88
16.37
11.57
17.64
19.94
TimeMixer
13.56
17.39
13.11
19.52
22.27
NMAE comparison between the proposed method and independent forecasting (%).
Method
Model
F1
F2
F3
F4
F5
FL + ML
Autoformer
6.30
8.56
6.20
9.38
11.25
Transformer
6.61
8.88
6.44
9.63
11.39
TimeMixer
7.00
9.60
6.73
9.75
11.35
Independent
Autoformer
10.14
13.02
9.73
13.38
17.57
Transformer
8.55
12.66
8.78
12.78
15.19
TimeMixer
10.67
13.78
10.28
14.54
17.02
Taking Autoformer (which yields the lowest errors under the proposed FL + ML setting) as an example, the average MAPE is reduced from 5.75% under independent forecasting to 3.76%, corresponding to an absolute reduction of 1.99 percentage points (with per-user reductions ranging from 1.63 to 2.17 percentage points). Similarly, NRMSE decreases from 16.81% to 12.17%, yielding an improvement of 4.64 percentage points (3.98–5.04 percentage points across users), while NMAE is reduced from 12.77% to 8.34%, i.e., a decrease of 4.43 percentage points (3.53–6.32 percentage points). Consistent gains are also observed for Transformer and TimeMixer when incorporating the proposed framework: relative to their independent counterparts, the average MAPE/NRMSE/NMAE decrease by 1.34/3.08/3.00 percentage points for Transformer and 2.00/4.77/4.37 percentage points for TimeMixer, respectively. These results demonstrate that under limited data availability and heterogeneous load patterns, federated collaboration improves global representation learning, while meta-learning further strengthens rapid cross-factory adaptation, leading to stable accuracy gains without sharing raw load data.
The multi-user forecasting curves shown in Figure 4 provide a direct visual comparison. For all five industrial users (F1–F5), the proposed method (Proposed, red curves) exhibits the closest alignment with the ground truth (True, black curves). In critical transition periods—such as the morning ramp-up from off-peak valleys to daytime loads and the subsequent decline—the proposed method more accurately tracks trend changes, effectively mitigating the amplitude bias and phase lag commonly observed in independent forecasting models at peak and valley regions. Moreover, during periods of pronounced load volatility (e.g., around peak loads of F1, F3, and F5), the proposed method produces smoother trajectories while retaining responsiveness to sharp variations, avoiding the local overshoot or underestimation seen in independently trained Autoformer, Transformer, and TimeMixer models. This behavior reflects enhanced robustness.
Load forecasting curves for various industrial users.
Grouped line charts compare predicted and actual electric load (MW) across five regions labeled F1 to F5, with time on the x-axis and multiple forecasting models shown in various colors for each subplot.
Further comparisons with hybrid baselines (e.g., ML + FL + Transformer and ML + FL + TimeMixer) indicate in Figure 5 that the proposed framework maintains a consistent advantage across users, confirming that the combination of federated knowledge sharing and meta-learning–based rapid adaptation jointly improves the modeling of heterogeneous industrial load patterns, leading to more reliable short-term forecasting under privacy constraints.
Radar chart showing the performance of different methods across different users.
Three radar charts compare six forecasting models—Proposed, ML+FL+Timemixer, Transformer, ML+FL+Transformer, Autoformer, and Timemixer—across metrics MAPE, NRMSE, and NMAE for features F1 to F5, with Timemixer consistently covering a larger area.Validation of rapid adaptation capability for newly connected users based on meta-learning
In the new-scenario, small-sample test for the newly connected user F6, Table 6 compares the prediction errors of three approaches under different backbone models: the proposed federated learning + meta-learning method (Proposed, ML + FL), conventional federated learning (FL), and local modeling based solely on small samples. The results show that the proposed method achieves the best and most stable performance across all three metrics.
Load forecasting error table for F6 (%).
Method
Model
MAPE
NRMSE
NMAE
Proposed (ML + FL)
Autoformer
3.85
9.36
7.22
Transformer
3.80
9.27
7.20
Timemixer
3.92
9.90
7.42
Federated
Autoformer
5.93
14.27
11.21
Transformer
5.28
13.13
10.04
Timemixer
6.16
14.94
11.69
Independent (few samples)
Autoformer
6.48
15.92
12.37
Transformer
5.78
13.69
10.92
Timemixer
6.96
16.64
13.21
Taking Autoformer as an example, the Proposed method achieves MAPE/NRMSE/NMAE values of 3.85%/9.36%/7.22%, respectively. Compared with conventional FL (5.93%/14.27%/11.21%), the errors are reduced by 2.08, 4.91, and 3.99 percentage points, respectively. Compared with local small-sample modeling (6.48%/15.92%/12.37%), the errors are further reduced by 2.63, 6.56, and 5.15 percentage points. Consistent conclusions are observed for Transformer and TimeMixer. Relative to conventional FL, MAPE is reduced by 1.48–2.24 percentage points, NRMSE by 3.86–5.04 percentage points, and NMAE by 2.84–4.27 percentage points. Relative to independent few-sample modeling, the Proposed method reduces MAPE by 1.98–3.04 percentage points, NRMSE by 4.42–6.74 percentage points, and NMAE by 3.72–5.79 percentage points. These results indicate that, after introducing meta-learning, the global model can leverage cross-factory collaborative knowledge to form a more transferable initialization, enabling rapid personalized adaptation under extremely limited data conditions for F6.
The load curve comparisons in Figure 6 further corroborate these findings. During multiple peak–valley transitions and ramp-up/ramp-down phases of F6, the proposed method exhibits closer alignment with the ground-truth load curve. It not only tracks overall trend variations more accurately, but also effectively suppresses common issues observed in conventional FL and small-sample modeling, such as peak underestimation, valley shifts, and amplified local fluctuations. In particular, during periods of rapid load increase or sudden drop, the proposed method responds more promptly with reduced phase lag, demonstrating stronger short-term dynamic modeling capability. Combined with the quantitative results in Table 6, it can be concluded that under privacy constraints where raw load data are not shared, the proposed federated collaboration + meta-learning–based rapid adaptation framework effectively mitigates the performance degradation caused by cold-start issues in newly connected users, achieving efficient transfer learning and robust forecasting in new access scenarios.
Load forecasting result curves of various prediction methods.
Line chart compares predicted and true values of electricity load in megawatts across multiple forecasting models over time in fifteen-minute intervals. The proposed model closely follows the true load values.Federated learning–based forecasting under differential privacy constraints
Under DP constraints, to validate the effectiveness of the proposed Differentiated Privacy Protection Weight (DP) strategy, it is compared with the Maximum Privacy Protection Weight (MP) scheme, in which the strictest noise level is imposed on all users. The results demonstrate that DP achieves superior accuracy across all users (F1–F5) and backbone models, as shown in Table 7–9.
MAPE of different prediction methods for each user (%).
Method
Model
F1
F2
F3
F4
F5
Proposed (DP)
Autoformer
3.61
3.74
3.69
3.93
3.85
Transformer
3.78
3.89
3.84
4.03
3.89
Timemixer
3.98
4.19
3.98
4.08
3.88
MP
Autoformer
5.02
4.95
5.08
5.06
4.77
Transformer
4.44
4.66
4.76
4.83
4.70
Timemixer
4.46
4.73
4.48
4.61
4.38
NRMSE of different prediction methods for each user (%).
Method
Model
F1
F2
F3
F4
F5
Proposed (DP)
Autoformer
8.50
11.70
8.36
14.63
17.66
Transformer
9.01
12.42
8.52
14.88
16.18
Timemixer
9.05
13.57
9.21
14.37
15.80
MP
Autoformer
11.12
14.52
10.90
17.28
19.52
Transformer
10.33
14.40
10.95
17.31
18.46
Timemixer
10.13
15.20
10.41
15.92
17.73
NMAE of different prediction methods for each user (%).
Method
Model
F1
F2
F3
F4
F5
Proposed (DP)
Autoformer
6.30
8.56
6.20
9.38
11.25
Transformer
6.61
8.88
6.44
9.63
11.39
Timemixer
7.00
9.60
6.73
9.75
11.35
MP
Autoformer
8.82
11.29
8.55
12.06
13.89
Transformer
7.82
10.66
7.99
11.52
13.76
Timemixer
7.85
10.84
7.56
11.01
12.81
Taking Autoformer as an example, the average MAPE under DP decreases from 4.976% (MP) to 3.764%, corresponding to a reduction of 1.21 percentage points (with per-user reductions ranging from 0.92 to 1.41 percentage points). The average NRMSE decreases from 14.668% to 12.170%, yielding an improvement of 2.50 percentage points, while the average NMAE decreases from 10.922% to 8.338%, corresponding to a reduction of 2.58 percentage points. Consistent trends are also observed for Transformer and TimeMixer: compared with MP, DP reduces MAPE by approximately 0.79 and 0.51 percentage points, NRMSE by about 2.09 and 1.48 percentage points, and NMAE by about 1.76 and 1.13 percentage points, respectively. These results indicate that differentiated noise allocation can effectively alleviate the accuracy degradation caused by uniformly strong noise, without compromising privacy protection requirements.
The error boxplots further confirm these findings from a distributional perspective in Figure 7. Compared with the MP strategy, which exhibits higher median errors and wider interquartile ranges (indicating larger dispersion) for multiple users, the DP strategy produces boxplots that are overall closer to the low-error region, with smaller interquartile ranges and fewer extreme values, reflecting improved stability and robustness. This is mainly because MP homogenizes all users’ privacy requirements to the highest level, forcing most users to inject excessive noise and thereby diluting the effective information during federated aggregation. In contrast, DP configures noise intensity according to users’ privacy levels, providing stronger protection for highly sensitive users while avoiding unnecessary perturbation for less sensitive ones. As a result, a more favorable trade-off between privacy strength and prediction accuracy is achieved during global aggregation, leading to consistent error reduction and variance convergence across users.
Box plots of prediction errors for different methods across various users.
Boxplot comparison of seven forecasting models across five datasets (F1 to F5) using three metrics: MAPE percentage, NRMSE percentage, and NMAE percentage. Models include Proposed, DP plus Transformer, DP plus Timemixer, MP plus Autoformer, MP plus Transformer, and MP plus Timemixer, each distinguished by color in the legend. Lower error metric values are visually favored by the Proposed model across all datasets and metrics.Conclusion
In short-term load forecasting for industrial users, prediction accuracy is often constrained by limited single-user data and complex load patterns, while cross-factory collaborative forecasting faces substantial privacy leakage risks due to the involvement of production-sensitive load data. To address these challenges, this paper proposes a privacy-preserving cross-scenario collaborative forecasting framework. Federated learning is employed to establish a secure collaborative training mechanism under the principle of “data available but not visible.” Meta-learning is further introduced to alleviate negative transfer caused by multi-factory heterogeneity and to enhance rapid adaptation capability. In addition, a differentiated noise-weight allocation strategy for DP budgets is designed to achieve an improved trade-off between prediction accuracy and privacy protection under diverse privacy requirements. Case studies demonstrate that the proposed framework exhibits consistent advantages across different users and backbone models.
Compared with traditional independent training, the proposed federated learning + meta-learning approach significantly improves forecasting accuracy. Taking Autoformer as an example, the average MAPE decreases from 5.754% to 3.764%, corresponding to a reduction of 1.99 percentage points.
The proposed method achieves rapid personalized adaptation with only a small amount of local data. For Autoformer, the MAPE for F6 is reduced to 3.85%, representing a decrease of 2.08 percentage points compared with conventional federated learning, and a decrease of 2.63 percentage points compared with local small-sample modeling.
The differentiated privacy noise-weight strategy effectively mitigates the accuracy degradation caused by uniformly strong noise when compared with the maximum privacy protection weight. For Autoformer, the average MAPE decreases from 4.976% to 3.764%, corresponding to a reduction of 1.21 percentage points.
In summary, the proposed approach provides a practical and deployable technical pathway for secure collaborative load forecasting for industrial users in power systems. By improving prediction accuracy and alleviating cold-start issues for newly connected users without exposing raw data, it offers more reliable load priors for day-ahead scheduling, load management and demand response, reserve capacity allocation, and operational risk control, thereby enhancing refined operation and coordinated control on the industrial load side.
Data availability statement
The data analyzed in this study is subject to the following licenses/restrictions: The authors do not have permission to share data. Requests to access these datasets should be directed to Maomao Ding, mmding8909_bjqr@163.com.
Author contributions
MD: Conceptualization, Writing – review and editing, Writing – original draft, Methodology. ZC: Investigation, Writing – review and editing, Software, Writing – original draft. BC: Formal Analysis, Writing – original draft, Data curation. JH: Resources, Writing – review and editing, Methodology, Investigation.
Conflict of interest
Authors MD, ZC, and BC were employed by State Grid Corporation of China. Author JH was employed by Beijing Tsingsoft Technology Co., Ltd.
Generative AI statement
The author(s) declared that generative AI was used in the creation of this manuscript. During the preparation of this manuscript, the author(s) used ChatGPT-4o for the purpose of polishing the English writing and language. The authors have reviewed and edited the output and take full responsibility for the content of this publication.
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Edited by:Jiaqi Shi, Shenyang Institute of Engineering, China
Reviewed by:Xiang Zhang, North China Electric Power University, China
Hongyang Jin, Shenyang Institute of Engineering, China