Hypergraphs provide a natural and expressive framework for modeling high-order relationships, enabling the representation of group-wise interactions beyond pairwise connections. While hypergraph neural networks (HNNs) have shown promise for learning on such structures, existing models often rely on shallow message passing and lack the ability to extract multiscale patterns. Framelet-based techniques offer a principled solution by decomposing signals into multiple frequency bands. However, most prior framelet systems, particularly Haar-type ones, are sensitive to node ordering and fail to ensure consistent representations under permutation, leading to instability in hypergraph learning. To address this, we propose Permutation Equivariant Framelet-based Hypergraph Neural Networks (PEF-HNN), a novel framework that integrates multiscale framelet analysis with permutation-consistent learning. We construct a new family of permutation equivariant Haar-type framelets specifically designed for hypergraphs, supported by theoretical analysis of their stability and decomposition properties. Built upon these framelets, PEF-HNN incorporates both low-pass and high-pass components across multiple scales into a unified neural architecture. Extensive experiments on nine benchmark datasets, including three homophilic and four heterophilic hypergraphs, as well as two real-world datasets for visual object classification, demonstrate the effectiveness of our approach, consistently outperforming existing HNN baselines and highlighting the advantages of permutation equivariant framelet design in hypergraph representation learning.

The out-of-distribution (OOD) detection on graph-structured data is crucial for deploying graph neural networks securely in open-world scenarios. However, existing methods have overlooked the prevalent scenario of multi-label classification in real-world applications. In this work, we investigate the unexplored issue of OOD detection within multi-label node classification tasks. We propose ML-GOOD, a simple yet sufficient approach that utilizes an energy function to gauge the OOD score for each label. We further develop a strategy for amalgamating multiple label energies, allowing for the comprehensive utilization of label information to tackle the primary challenges encountered in multi-label scenarios. Extensive experimentation conducted on seven diverse sets of real-world multi-label graph datasets, encompassing cross-domain scenarios. The results show that the AUROC of ML-GOOD is improved by 5.26% in intra-domain and 6.54% in cross-domain compared to the previous methods. These empirical validations not only affirm the robustness of our methodology but also illuminate new avenues for further exploration within this burgeoning field of research.

Hypergraphs provide a flexible framework for modeling high-order (complex) interactions among multiple entities, extending beyond traditional pairwise correlations in graph structures. However, deep hypergraph neural networks (HGNNs) often face the challenge of oversmoothing with increasing depth, similar to issues in graph neural networks (GNNs). While oversmoothing in GNNs has been extensively studied, its implications in relation to hypergraphs are less explored. This paper addresses this gap by first theoretically exploring the reasons behind oversmoothing in deep HGNNs. Our novel insights suggest that a spectral-based hypergraph convolution, equipped with both low-pass and high-pass filters, can potentially mitigate these effects. Motivated by these findings, we introduce FrameHGNN, a framework that utilizes framelet-based hypergraph convolutions integrating tight framelet transforms with both low-pass and high-pass components, as well as the commonly used strategies in designing deep GNN architecture: initial residual and identity mappings. The experiment results on diverse benchmark datasets demonstrate that FrameHGNN outperforms several state-of-the-art models, effectively reducing oversmoothing while improving predictive accuracy. Our contributions not only advance the theoretical understanding of deep hypergraph learning but also provide a practical spectral-based approach for HGNNs, emphasizing the design of multifrequency channels.

Hypergraph neural networks (HNNs) have shown promise in handling tasks characterized by high-order correlations, achieving notable success across various applications. However, there has been limited focus on heterophilic hypergraph learning (HHL), in contrast to the increasing attention given to graph neural networks designed for graphs exhibiting heterophily. This paper aims to pave the way for HHL by addressing key gaps from multiple perspectives: measurement, dataset diversity, and baseline model development. First, we introduce metrics to quantify heterophily in hypergraphs, providing a numerical basis for assessing the homophily/heterophily ratio. Second, we develop diverse benchmark datasets across various real-world scenarios, facilitating comprehensive evaluations of existing HNNs and advancing research in HHL. Additionally, as a novel baseline model, we propose HyperUFG, a framelet-based HNN integrating both low-pass and high-pass filters. Extensive experiments conducted on synthetic and benchmark datasets highlight the challenges current HNNs face with heterophilic hypergraphs, while showcasing that HyperUFG performs competitively and often outperforms many existing models in such scenarios. Overall, our study underscores the urgent need for further exploration and development in this emerging field, with the potential to inspire and guide future research in HHL.

With the growing adoption of Hypergraph Neural Networks (HNNs) to model higher-order relationships in complex data, concerns about their security and robustness have become increasingly important. However, current security research often overlooks the unique structural characteristics of hypergraph models when developing adversarial attack and defense strategies. To address this gap, we demonstrate that hypergraphs are particularly vulnerable to node injection attacks, which align closely with real-world applications. Through empirical analysis, we develop a relatively unnoticeable attack approach by monitoring changes in homophily and leveraging this self-regulating property to enhance stealth. Building on these insights, we introduce HyperNear, i.e., $\underline{N}$ode inj$\underline{E}$ction $\underline{A}$ttacks on hype$\underline{R}$graph neural networks, the first node injection attack framework specifically tailored for HNNs. HyperNear integrates homophily-preserving strategies to optimize both stealth and attack effectiveness. Extensive experiments show that HyperNear achieves excellent performance and generalization, marking the first comprehensive study of injection attacks on hypergraphs. Our code is available at https://github.com/ca1man-2022/HyperNear.

Diffusion models, known for their generative ability, have recently been adapted to time series analysis. Most pioneering works rely on the standard isotropic diffusion, treating each time step and the entire frequency spectrum identically. However, it may not be suitable for time series, which often have more informative low-frequency components. We empirically found that direct application of standard diffusion to time series may cause gradient contradiction during training, due to the rapid decrease of low-frequency information in the diffusion process. To this end, we proposed a novel time series diffusion model, MA-TSD, which utilizes the moving average, a natural low-frequency filter, as the forward transition. Its backward process is accelerable like DDIM and can be further considered a time series super-resolution. Our experiments on various datasets demonstrated MA-TSD’s superior performance in time series forecasting and super-resolution tasks.

Heterophilic graph neural networks (GNNs) have gained prominence for their ability to learn effective representations in graphs with diverse, attribute-aware relationships. While existing methods leverage attribute inference during message passing to improve performance, they often struggle with challenging heterophilic graphs. This is due to edge distribution shifts introduced by diverse connection patterns, which blur attribute distinctions and undermine message-passing stability. This paper introduces H₂OGNN, a novel framework that reframes edge attribute inference as an out-of-distribution (OOD) detection problem. H₂OGNN introduces a simple yet effective symbolic energy regularization approach for OOD learning, ensuring robust classification boundaries between homophilic and heterophilic edge attributes. This design significantly improves the stability and reliability of GNNs across diverse connectivity patterns. Through theoretical analysis, we show that H₂OGNN addresses the graph denoising problem by going beyond feature smoothing, offering deeper insights into how precise edge attribute identification boosts model performance. Extensive experiments on nine benchmark datasets demonstrate that H₂OGNN not only achieves state-of-the-art performance but also consistently outperforms other heterophilic GNN frameworks, particularly on datasets with high heterophily.

The coordinated operation of grid-interactive buildings contributes to creating a more resilient and reliable power grid. However, existing studies fail to identify the demand changes of each building resulting from their coordinated participation in providing grid services, which affects the economic compensation of each building and its willingness to coordinate. In this article, we investigate an optimal coordinated operation problem for grid-interactive residential buildings (GRBs) while considering generation capacity services and economic compensation for participating GRBs. Specifically, we first formulate two optimization problems to capture the different objectives of GRBs during non-service periods and grid-service periods, respectively. Then, we develop a physically consistent neural network (PCNN)-assisted hierarchical model predictive control (HMPC)-based GRB energy management algorithm to solve the optimization problem during non-service periods. Next, we propose a coordinated operation algorithm to solve the optimization problem during grid-service periods based on PCNN-assisted HMPC and rule-assisted binary search. By comparing the initial solutions from the proposed energy management algorithm with the final solutions generated by the proposed coordination algorithm, the demand changes of each GRB during service periods can be identified. Simulation results indicate that the proposed coordination algorithm achieves up to 37.8114% lower energy costs and 82.1459% better grid service performance than benchmarks while maintaining high thermal comfort. Note to Practitioners—Buildings with distributed energy resources (e.g., solar generation, energy storage) and flexible loads (e.g., heating, ventilation, and air conditioning (HVAC) systems) have significant potential to provide grid services, such as voltage support, frequency regulation, and power peak reduction. Since a single residential building contributes minimally to service quality, multi-building coordination through a trusted third party is necessary. However, since participation in providing grid services may affect occupant comfort and building energy costs, the demand change of each residential building should be identified so that the corresponding economic compensation can be calculated, which is a key factor for the successful deployment of such grid-interactive residential buildings (GRBs). To this end, we develop an optimal energy management algorithm for each residential building during non-service periods, which aims to minimize building energy cost while maintaining high occupant comfort. Based on the developed energy management algorithm, a coordinated operation algorithm for service periods is further proposed to limit the peak demand below a value predetermined by the system operator. By comparing the initial decisions from the energy management algorithm with the final decisions generated by the proposed coordination algorithm, we can identify the demand change of each building during service periods. Numerical results demonstrate that the proposed coordinated operation algorithm can help participating GRBs reduce energy costs and enhance service performance for power grids, with negligible sacrifice to occupant comfort.

Time series forecasting is a critical and challenging task in practical application. Recent advancements in pre-trained foundation models for time series forecasting have gained significant interest. However, current methods often overlook the multi-scale nature of time series, which is essential for accurate forecasting. To address this, we propose HiMTM, a hierarchical multi-scale masked time series modeling with self-distillation for long-term forecasting. HiMTM integrates four key components: (1) hierarchical multi-scale transformer (HMT) to capture temporal information at different scales; (2) decoupled encoder-decoder (DED) that directs the encoder towards feature extraction while the decoder focuses on pretext tasks; (3) hierarchical self-distillation (HSD) for multi-stage feature-level supervision signals during pre-training; and (4) cross-scale attention fine-tuning (CSA-FT) to capture dependencies between different scales for downstream tasks. These components collectively enhance multi-scale feature extraction in masked time series modeling, improving forecasting accuracy. Extensive experiments on seven mainstream datasets show that HiMTM surpasses state-of-the-art self-supervised and end-to-end learning methods by a considerable margin of 3.16-68.54%. Additionally, HiMTM outperforms the latest robust self-supervised learning method, PatchTST, in cross-domain forecasting by a significant margin of 2.3%. The effectiveness of HiMTM is further demonstrated through its application in natural gas demand forecasting.

Explaining multivariate time series is a compound challenge, as it requires identifying important locations in the time series and matching complex temporal patterns. Although previous saliency-based methods addressed the challenges, their perturbation may not alleviate the distribution shift issue, which is inevitable especially in heterogeneous samples. We present ContraLSP, a locally sparse model that introduces counterfactual samples to build uninformative perturbations but keeps distribution using contrastive learning. Furthermore, we incorporate sample-specific sparse gates to generate more binary-skewed and smooth masks, which easily integrate temporal trends and select the salient features parsimoniously. Empirical studies on both synthetic and real-world datasets show that ContraLSP outperforms state-of-the-art models, demonstrating a substantial improvement in explanation quality for time series data. The source code is available at \url{https://github.com/zichuan-liu/ContraLSP}.

Time series analysis is widely used in many fields such as power energy, economics, and transportation, including different tasks such as forecasting, anomaly detection, classification, etc. Missing values are widely observed in these tasks, and often leading to unpredictable negative effects on existing methods, hindering their further application. In response to this situation, existing time series imputation methods mainly focus on restoring sequences based on their data characteristics, while ignoring the performance of the restored sequences in downstream tasks. Considering different requirements of downstream tasks (e.g., forecasting), this paper proposes an efficient downstream task-oriented time series imputation evaluation approach. By combining time series imputation with neural network models used for downstream tasks, the gain of different imputation strategies on downstream tasks is estimated without retraining, and the most favorable imputation value for downstream tasks is given by combining different imputation strategies according to the estimated gain.

Message passing (MP) is crucial for effective graph neural networks (GNNs). Most local message-passing schemes have been shown to underperform on heterophily graphs due to the perturbation of updated representations caused by local redundant heterophily information. However, our experiment findings indicate that the distribution of heterophily information during MP can be disrupted by disentangling local neighborhoods. This finding can be applied to other GNNs, improving their performance on heterophily graphs in a more flexible manner compared to most heterophily GNNs with complex designs. This article proposes a new type of simple message-passing neural network called Flow2GNN. It uses a two-way flow message-passing scheme to enhance the ability of GNNs by disentangling and redistributing heterophily information in the topology space and the attribute space. Our proposed message-passing scheme consists of two steps in topology space and attribute space. First, we introduce a new disentangled operator with binary elements that disentangle topology information in-flow and out-flow between connected nodes. Second, we use an adaptive aggregation model that adjusts the flow amount between homophily and heterophily attribute information. Furthermore, we rigorously prove that disentangling in message-passing can reduce the generalization gap, offering a deeper understanding of how our model enhances other GNNs. The extensive experiment results show that the proposed model, Flow2GNN, not only outperforms state-of-the-art GNNs, but also helps improve the performance of other commonly used GNNs on heterophily graphs, including GCN, GAT, GCNII, and H2GCN, specifically for GCN, with up to a 25.88% improvement on the Wisconsin dataset.

The proliferation of novel industrial applications at the wireless edge, such as smart grids and vehicle networks, demands the advancement of cyber-physical systems (CPSs). The performance of CPSs is closely linked to the last-mile wireless communication networks, which often become bottlenecks due to their inherent limited resources. Current CPS operations often treat wireless communication networks as unpredictable and uncontrollable variables, ignoring the potential adaptability of wireless networks, which results in inefficient and overly conservative CPS operations. Meanwhile, current wireless communications often focus more on throughput and other transmission-related metrics instead of CPS goals. In this study, we introduce the framework of goal-oriented wireless communication resource allocations, accounting for the semantics and significance of data for CPS operation goals. This guarantees optimal CPS performance from a cybernetic standpoint. We formulate a bandwidth allocation problem aimed at maximizing the information utility gain of transmitted data brought to CPS operation goals. Since the goal-oriented bandwidth allocation problem is a large-scale combinational problem, we propose a divide-and-conquer and greedy solution algorithm. The information utility gain is first approximately decomposed into marginal utility information gains and computed in a parallel manner. Subsequently, the bandwidth allocation problem is reformulated as a knapsack problem, which can be further solved greedily with a guaranteed sub-optimality gap. We further demonstrate how our proposed goal-oriented bandwidth allocation algorithm can be applied in four potential CPS applications, including data-driven decision-making, edge learning, federated learning, and distributed optimization. Through simulations, we confirm the effectiveness of our proposed goal-oriented bandwidth allocation framework in meeting CPS goals.

Accurate prediction of electric load is crucial in power grid planning and management. In this paper, we solve the electric load forecasting problem under extreme events such as scorching heats. One challenge for accurate forecasting is the lack of training samples under extreme conditions. Also load usually changes dramatically in these extreme conditions, which calls for interpretable model to make better decisions. In this paper, we propose a novel forecasting framework, named Self-adaptive Decomposed Interpretable framework (SaDI), which ensembles long-term trend, short-term trend, and period modelings to capture temporal characteristics in different components. The external variable triggered loss is proposed for the imbalanced learning under extreme events. Furthermore, Generalized Additive Model (GAM) is employed in the framework for desirable interpretability. The experiments on both Central China electric load and public energy meters from buildings show that the proposed SaDI framework achieves average 22.14% improvement compared with the current state-of- the-art algorithms in forecasting under extreme events in terms of daily mean of normalized RMSE. Code, Public datasets, and Appendix are available at: https://doi.org/10.24433/CO.9696980.v1.

We use symmetric cryptography for secure communications with resource-limited smart grid control devices. We propose the novel idea of using the unmanned aerial vehicle (UAV) as a physical courier to carry secret key bits generated at the control center to remote control devices. While distributing secret keys, the UAV may be attacked in an attempt to sabotage secure communications. The UAV may fly a longer route to avoid attackers but it has a limited onboard battery capacity and must visit each device within a time limit to prevent a secret key deficit. We formulate an optimization to find the lowest attack risk flight route for secret key distribution while taking into consideration key deficiency and battery capacity. Such optimization is NP-hard. We further propose an efficient algorithm to find a low-risk flight route in less than a second, even for a large power grid system with more than a hundred nodes. For large systems, we modify the proposed algorithm to find a low-risk flight route as well as to determine the required UAV flight duration and battery size. Extensive simulation results confirm the superiority of the proposed algorithm against a baseline scheme that is ignorant of attackers.

With increasing distributed energy resource integration, future power and energy systems will be more decentralized using advanced Internet of Things (IoT) technologies. Integrated energy systems (IESs) boost the whole energy efficiency by coordinating multiregional energy resources and networks. However, distributed coordination of the IES requires different subregions or energy hubs (EHs) to share their sensitive information (e.g., energy demands and operation status) explicitly, which poses serious privacy leakage. To this end, secure multiparty computation (SMPC) is innovatively introduced to the distributed optimization of the IES in this article. First, the standardized modeling of multiple interconnected EHs with the linearized network models is formulated to analyze the IES’s inherent energy and information interaction comprehensively. Then, a privacy-preserving distributed optimal energy flow algorithm is proposed by combining the Paillier Cryptosystem mechanism with the alternating direction multiplier method (ADMM). Theoretical analysis proves the proposed method is convergent without sharing sensitive information in plaintext. Numerical experiments on a three-subregions IES validate that the proposed method has better convergence performance than the differential privacy-based method. Results show that the maximum relative error of the distributed optimal solutions with various step sizes is no more than 0.072% compared with the centralized method.

Timely and high-density air quality monitoring is essential for the development of future smart cities. The images captured from widely deployed stationary-cameras can be transferred quickly via the Internet of Things (IoT) to facilitate ambient pollution estimation anytime anywhere. Image-based air pollution estimation is normally formulated as a supervised learning problem, relying on an extended number of image samples. However, individual stationary-cameras can offer only very limited samples and scenes, while locally trained estimation models can easily overfit. A global method was proposed to address this challenge. The global model was trained via images captured from different cameras. However, such a model is less effective in extracting local features from scenes. A personalized method is therefore proposed to improve not only the generalization of the estimation model but also to preserve the local characteristics of individual cameras. Our personalized method consists of a two-stage architecture: 1) images from different cameras are used to train the global estimation model to avoid overfitting due to fixed scenes and small sample size and 2) the global model is further refined by images captured from individual cameras separately for adapting local characteristics. To evaluate our proposed personalized method, a large data set was constructed, based on stationary-camera-taken images captured in Hong Kong, consisting of different pollution measurements, including PM2.5, PM10, NO2, and O3. As compared to the local model, our proposed personalized model has reduced average MAE by 5.68% and average SMAPE by 6.82%, and improved average$r$by 4.69%.

An increasing number of distributed energy resources (DERs), such as rooftop photovoltaic (PV), electric vehicles (EVs), and distributed energy storage, are being integrated into the distribution systems. The rise of DERs has come hand-in-hand with large amounts of data generated and explosive growth in data collection, communication, and control devices. In addition, a massive number of consumers are involved in the interaction with the power grid to provide flexibility. Electricity consumers, power networks, and communication networks are three main parts of the distribution systems, which are deeply coupled. In this sense, smart distribution systems can be essentially viewed as cyber–physical–social systems. So far, extensive works have been conducted on the intersection of cyber, physical, and social aspects in distribution systems. These works involve two or three of the cyber, physical, and social aspects. Having a better understanding of how the three aspects are coupled can help to better model, monitor, control, and operate future smart distribution systems. In this regard, this article provides a comprehensive review of the coupling relationships among the cyber, physical, and social aspects of distribution systems. Remarkably, several emerging topics that challenge future cyber–physical–social distribution systems, including applications of 5G communication, the impact of COVID-19, and data privacy issues, are discussed. This article also envisions several future research directions or challenges regarding cyber–physical–social distribution systems.

The operations of base stations (BSs) contribute most of the energy consumption in the cellular wireless networks. Powering BSs by distributed energy resources (DERs), such as photovoltaic (PV) and energy storage, is an effective way to reduce on-grid power consumption and build green wireless networks. Optimal energy management of BSs helps to reduce electricity bills for the wireless network and provides flexibility to the power networks. This article proposes the concept of spatial–temporal energy management (ST-EM) for the energy management of BSs. On the one hand, the BSs manage their power consumption according to the real-time prices; on the other hand, the BSs adjust the user associations and change their power consumption according to the price differences among different BSs. The ST-EM of BSs is formulated as a large-scale mixed-integer nonlinear programming (MINLP) problem, which is proven to be NP-hard. We propose a heuristic approach to search for one suboptimal solution by decomposing the original problem into an energy network optimization subproblem and a communication network optimization subproblem. The two subproblems are solved alternatively until convergence. Numerical experiments are conducted to verify the effectiveness of our proposed method.

Scenario generation is a fundamental and crucial tool for decision-making in power systems with high-penetration renewables. Based on big historical data, in this article, a novel federated deep generative learning framework, called Fed-LSGAN, is proposed by integrating federated learning and least square generative adversarial networks (LSGANs) for renewable scenario generation. Specifically, federated learning learns a shared global model in a central server from renewable sites at network edges, which enables the Fed-LSGAN to generate scenarios in a privacy-preserving manner without sacrificing the generation quality by transferring model parameters, rather than all data. Meanwhile, the LSGANs-based deep generative model generates scenarios that conform to the distribution of historical data through fully capturing the spatial-temporal characteristics of renewable powers, which leverages the least squares loss function to improve the training stability and generation quality. The simulation results demonstrate that the proposal manages to generate high-quality renewable scenarios and outperforms the state-of-the-art centralized methods. Besides, an experiment with different federated learning settings is designed and conducted to verify the robustness of our method.

Multienergy systems (MESs) are able to unlock the energy system flexibility using the coupling across multiple energy sectors. Such coupling contributes to improving the overall energy efficiency and promoting the accommodation of renewable energy. Among a wide range of literature, this article provides a perspective of network analytics on how to model, optimize, and conduct low-carbon analysis on MESs. The energy sector coupling involves different levels, for example, from a single building to nationwide. In this article, we categorize multienergy networks into two levels, that is, the district level that covers a relatively small area such as a campus or a community, where the energy conversion and utilization is the major focus, and the multiregion level that covers a relatively large area such as a big city, a province, or the whole country, where the energy transmission is the major concern. We first review the state-of-the-art multienergy networks standardized modeling approaches including: 1) energy hub (EH) model for district level energy networks; 2) network models, including power, heat, and gas steady-state and dynamic network models, for multiregion level energy networks; and 3) load models, including electricity, heat, and gas load forecasting models. Second, we explore the planning and operation methods for both district level and multiregion level energy networks. Third, we introduce a special technique named the carbon emission flow (CEF) model that is able to calculate the equivalent CO2 emission associated with the energy flows in multienergy networks. We also demonstrate how the technique can help multienergy networks planning and operation toward a low carbon society. Finally, we envision several further key research topics in the field of multienergy networks.

The integration of multienergy systems (MES) provides an important opportunity to increase the technical, economic, and environmental performance of the overall system compared with separately operated energy systems. The operation of district MES encounters numerous uncertainties such as fluctuations in energy prices as well as loads in different energy forms. In this paper, we formulate a probabilistic optimal energy flow (P-OEF) model of a district MES into a linear program (LP) problem based on the linear energy hub model. We propose an improved multiparametric LP (MPLP) based online dictionary-learning approach to efficiently calculate the P-OEF problem. We then derive a generalized critical region theory of the MPLP problem to jointly consider the change of parameters in terms of both the objective function and the constraints of P-OEF in MES. Finally, we explore the physical significance of the MPLP's critical region in the P-OEF problem. We perform a case study to verify the validity and effectiveness of the proposed approach compared to the traditional Monte Carlo approach for the P-OEF of the MES.

The two-way flow of information and energy is an important feature of the Energy Internet. Data analytics is a powerful tool in the information flow that aims to solve practical problems using data mining techniques. As the problem of electricity thefts via tampering with smart meters continues to increase, the abnormal behaviors of thefts become more diversified and more difficult to detect. Thus, a data analytics method for detecting various types of electricity thefts is required. However, the existing methods either require a labeled dataset or additional system information, which is difficult to obtain in reality or have poor detection accuracy. In this paper, we combine two novel data mining techniques to solve the problem. One technique is the maximum information coefficient (MIC), which can find the correlations between the nontechnical loss and a certain electricity behavior of the consumer. MIC can be used to precisely detect thefts that appear normal in shapes. The other technique is the clustering technique by fast search and find of density peaks (CFSFDP). CFSFDP finds the abnormal users among thousands of load profiles, making it quite suitable for detecting electricity thefts with arbitrary shapes. Next, a framework for combining the advantages of the two techniques is proposed. Numerical experiments on the Irish smart meter dataset are conducted to show the good performance of the combined method.