Simulation of Malfunctions in Home Appliances’ Power Consumption

Abstract

Predicting errors in home appliances is crucial for maintaining the reliability and efficiency of smart homes. However, there is a significant lack of such data on appliance malfunctions that can be used in developing effective anomaly detection models. This research paper presents a novel approach for simulating errors of heterogeneous home appliance power consumption patterns. The proposed model takes normal consumption patterns as input and employs advanced algorithms to produce labeled anomalies, categorizing them based on the severity of malfunctions. One of the main objectives of this research involves developing models that can accurately reproduce anomaly power consumption patterns, highlighting anomalies related to major, minor, and specific malfunctions. The resulting dataset may serve as a valuable resource for training algorithms specifically tailored to detect and diagnose these errors in real-world scenarios. The outcomes of this research contribute significantly to the field of anomaly detection in smart home environments. The simulated datasets facilitate the development of predictive maintenance strategies, allowing for early detection and mitigation of appliance malfunctions. This proactive approach not only improves the reliability and lifespan of home appliances but also enhances energy efficiency, thereby reducing operational costs and environmental impact.

1. Introduction

A significant proportion of residential energy consumption is spent on activities such as cooking, washing clothes, and heating water. Specifically, space heating represents about 65% of the total energy consumption in households, while water heating accounts for approximately 14%, and cooking for approximately 6%. Other household activities and appliances consume another 13% [1]. However, residents do not realize the degree to which the actual energy consumption of their home appliances contributes to the total energy consumption. As a result, this lack of awareness can lead to an increase in energy usage by up to a third [2]. Furthermore, malfunctions in home appliances can result in a significant increase in energy consumption while remaining unnoticed by residents, as the home appliance continues to operate “normally” [3]. These unnoticed errors can lead to an increase in energy usage of up to 75% in certain cases, significantly affecting overall energy efficiency and costs [4]. Implementing effective anomaly detection systems can help identify these inefficiencies and malfunctions early, reducing unnecessary energy consumption while improving overall energy management in households.

Anomaly detection techniques can identify potential errors or anomalies in home appliance energy usage patterns, indicating a possible malfunction. When accurate, such methods can also be used for predictive maintenance. These anomaly detection techniques detect either anomaly points or patterns. As an example, support vector machines (SVM), may be employed to distinguish normal and abnormal usage patterns in data related to energy consumption [5]. Moreover, deep learning (DL) models are effective in recognizing intricate patterns from large datasets, making neural networks useful for identifying small variations in energy consumption over time [6]. The k-Nearest Neighbors (kNN) algorithm analyzes an appliance’s consumption compared to that of its closest neighbors to recognize outliers in datasets [7]. Likewise, time-series analysis is used to observe trends in energy use over time and identify variations that demonstrate problems [8]. By recreating incoming data and comparing it to the original, autoencoders are used to learn a compressed representation of typical energy usage patterns and spot abnormalities [9]. By identifying and mitigating malfunctions before they occur, adopting these strategies can significantly improve the reliability and effectiveness of household appliances.

However, implementing these techniques in real-time and real-world scenarios is challenging. Home appliance energy usage patterns are numerous and diverse, even for the same appliance’s model, due to varying climates and/or usage habits. Moreover, for most of the methods, there is a critical need for historical consumption data to accurately identify the anomalies. In addition, labeled data with anomalies or specific malfunctions is necessary to effectively monitor and detect errors. Numerous other challenges exist in applying these techniques, such as data variability, the need for large datasets, and the complexity of real-time processing. The wide range of home appliances and potential malfunctions complicate the detection of errors and anomalies. Furthermore, there is limited availability of public datasets to train supervised models, which limits the development of accurate anomaly detection systems. This lack of data is a significant barrier, as supervised models rely on large, labeled datasets to learn the difference between normal and abnormal behavior. Therefore, addressing these challenges requires substantial effort in data collection, labeling, and the development of robust models capable of handling diverse and dynamic patterns of appliance usage.

Within this context, this paper suggests a method for simulating errors for common home appliances such as fridges, washing machines, dryers, dishwashers, and water heaters. This method is designed to be general, allowing it to be applied to all types of appliances and patterns. It can simulate and produce relevant errors, including minor, major, and specific faults. This simulation method is crucial for training and testing anomaly detection methods, as well as for applying predictive maintenance techniques. By simulating such a wide range of errors, this approach provides comprehensive datasets that can be used to enhance the accuracy and robustness of anomaly detection systems. It allows the creation of realistic scenarios in which appliances fail in various ways while preparing detection algorithms to recognize and respond to actual malfunctions effectively. Additionally, the ability to simulate different types of errors supports the development of predictive maintenance strategies, which can significantly improve the reliability and lifespan of home appliances.

In general, this method contributes significantly to enhancing the overall energy efficiency of households, ensuring that home appliances function correctly and sustainably. The main novelties are:

• Novel and adaptable method for simulating errors across a wide range of home appliances. This method can be applied to various appliances and usage patterns, generating comprehensive datasets for anomaly detection while being generally applied (cross-appliance applicability).
• Simulation of a broad spectrum of errors, including minor, major, and specific malfunctions. This provides detailed and realistic scenarios crucial for the development and testing of robust anomaly detection and predictive maintenance algorithms.
• Facilitation of efficient training and testing of anomaly detection technologies and predictive maintenance strategies by providing comprehensive datasets that reflect realistic appliance usage and error patterns.
• Integration of anomalies seamlessly into normal consumption patterns, ensuring the resulting data is realistic and representative of actual malfunctions, thereby improving the reliability of detection algorithms.

The remainder of this paper is structured as follows: Section 2 presents the current methodologies and applications of simulations in the energy sector and the techniques used to detect anomalies in the power consumption patterns of home appliances. Section 3 includes an extensive analysis of the energy usage patterns of home appliances as well as the algorithms and techniques used to simulate error patterns in the power consumption data. Simulated datasets and the analyses of real and anomalous patterns for each home appliance are presented in Section 4. Finally, in the last section, Section 5, the paper summarizes the key findings of the research.

2. Literature Review

In this section, the most recent literature related to this research topic is addressed. A search on Google Scholar including the keywords “simulation, errors, home appliance, consumption” reveals nearly 32,000 publications. Despite this number of publications, only the top two results are directly relevant to this paper: “Closing the Gap Between Simulation and Measured Energy Use in Home Archetypes” [10] and “A Scalable Real-Time Non-Intrusive Load Monitoring System for the Estimation of Household Appliance Power Consumption” [11]. However, these papers discuss the simulation of energy consumption and not the simulation of error anomalies. To broaden the research scope, a more general search using “Simulation energy” yielded approximately 6,690,000 publications. The differences in results indicate that while there is a plethora of research for the energy sector, there are far fewer tailored simulations. Specifically, it is observed that there is a significant gap in the literature concerning techniques specifically designed for simulating home appliance errors. Moreover, to the best of the authors’ knowledge, and the Google search, there are no published papers on the simulation of anomalies in energy consumption of home appliance errors. Based on this criterion, the literature review is divided into two sections: Section 2.1 presents existing models for simulations in the energy sector, providing a detailed analysis of methodologies and applications. Following that, Section 2.2 covers the techniques for anomaly detection, highlighting the current methodologies specifically used for detecting anomalies in the power consumption patterns of home appliances.

2.1. Existing Models for Simulation in the Energy Sector

Existing models for simulating consumption patterns often employ statistical distributions to predict power usage with high accuracy. For example, methodologies using the Kolmogorov–Smirnov test and Pearson’s X2 test have forecasted power consumption of household appliances with maximum errors as low as 3.9% for fridges and 4.27% for lighting, and even lower for washing machines, dishwashers (less than 1%), and dryers (about 1.17%) [12]. Despite their accuracy, these models are often limited to specific home environments and cannot replicate abnormalities across diverse settings. As there are different types of home appliances, more adaptable modeling techniques are needed to accommodate different household configurations. Additionally, predictive models using machine learning techniques such as multiple linear regression, random forest regressor, and support vector machines have been developed to enhance the accuracy of energy consumption forecasts, demonstrating significant potential in improving residential energy management systems [13]. Another approach involves modeling energy consumption profiles of smart home appliances, tailored to specific environmental conditions. Et-Tolba et al., developed models for Moroccan conditions that consider user preferences and environmental factors to enhance energy efficiency and reduce consumption costs [14]. These models are often created and simulated using platforms like Matlab/Simulink (2023b) [15]. Similarly, Sánchez-Cervantes et al. introduced a Big-Data-and-machine-learning-based smart home energy management system (HEMS-IoT) designed to learn user behaviors and energy consumption patterns. This system employs the J48 machine learning algorithm to classify energy consumption and generate energy-saving recommendations based on user preferences, contributing to home comfort, safety, and energy efficiency [16]. The significant impact of consumer behavior and ambient factors on energy consumption is evident in these simulations, offering valuable insights into customizing energy consumption profiles to reflect actual usage scenarios and user needs.

L. Diao et al., examine the simulation of energy consumption in residential buildings with a focus on occupant behavior and its impact on energy usage [17]. Using data from the American Time Use Survey (ATUS), the authors applied k-means clustering methods to identify and classify occupant behavior patterns. The proposed model integrates these behaviors into a bottom-up simulation model to estimate energy consumption. It is highlighted that the proposed model offers more accurate and reliable predictions compared to the standardized schedules of the American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE), indicating the need for personalized simulation models that take into account actual occupant behaviors. Similarly, Chen et al., (2021) developed a stochastic simulator integrated with a building stock model to more accurately predict residential energy loads. This model uses clustering techniques and Markov chain simulations to reflect the heterogeneity of occupant behavior, enhancing the reliability of energy consumption forecasts for the community on national scales [18].

Another relevant work in this context by A. Dimara et al., discusses the application of simulation in building energy systems, encompassing computational, mathematical, and machine learning models to represent a building’s physical characteristics and operational strategies [19]. This technique supports various tasks throughout the renovation life cycle, such as decision-making integration, design scenario evaluation, real-time monitoring and diagnostics for energy management, and the incorporation of virtual reality for digital design and operation experiences. Despite its established role in building performance, significant challenges remain, particularly in integrating data from fragmented systems and modeling human–building interactions. This work defines the domain of building performance simulation, highlighting key use cases, prevalent simulation tools, and implementation challenges. Finally, Zhao et al., develop a practical data-mining approach using office appliance power consumption data to model occupant behavior and schedules for building energy simulations [20]. The study demonstrates that accurately predicting occupant behavior can significantly impact HVAC energy consumption in buildings, especially in different climate zones. This approach emphasizes the importance of integrating occupant behavior modeling into energy simulations to enhance building performance and energy efficiency. Similarly, Azar and Menassa present a comprehensive framework that combines data-mining techniques with building information modeling to predict occupant behavior and energy use in commercial buildings. Their approach highlights the role of detailed occupant behavior modeling in improving the accuracy of energy simulations and optimizing building performance [21].

2.2. Techniques for Anomaly Detection

In the context of detecting anomalies in the power consumption of home appliances, researchers focus on identifying unusual patterns that may indicate potential appliance failures or inefficient energy use. One notable approach is the ensemble anomaly detection (EAD) and collective contextual anomaly detection using sliding window (CCAD-SW) frameworks developed by Araya et al., (2017) [22]. The CCAD-SW framework enhances detection performance by identifying abnormal consumption patterns through the use of overlapping sliding windows, thereby recognizing collective contextual anomalies. Additionally, the EAD framework employs majority voting to merge different anomaly detection classifiers, which improves sensitivity and reduces false alarm rates. Compared to individual classifiers, the EAD framework significantly enhances anomaly detection performance.

Furthermore, closely related to this paper, another approach was introduced by Qian et al. (2024) [23]. Using the DBSCAN method, a gray-relation-projection-based clustering learning strategy was developed to detect abnormal patterns in power use. By grouping users with comparable consumption habits and detecting anomalies within these groups, this technique makes it possible to spot unusual usage patterns that might point to appliance problems. This technique is very accurate and resilient, as shown by PR-AUC values of 0.96 and AUC values of 0.911, which make it a useful tool for simulating and spotting possible problems in home appliances. Additionally, Pan et al., (2022) propose a high-dimensional energy consumption anomaly detection method based on deep learning. This method utilizes high-dimensional energy data to predict and detect abnormal electricity usage in real-time, demonstrating its effectiveness in reducing energy waste and improving energy efficiency in buildings [24].

Priyadarshini et al., (2022) proposed a reliable approach for evaluating time series data from smart meters using several machine learning algorithms, including ARIMA, SARIMA, LSTM, Prophet, LightGBM, and VAR [25]. Their method uses the ChangeFinder algorithm, which recognizes shifts in consumption trends, to identify unusual usage patterns that can point to appliance issues. The outstanding MAE, MSE, and RMSE values of the ARIMA model demonstrate its great performance, highlighting its dependability in forecasting and simulating home appliance problems. This thorough process ensures that potential issues are quickly found and fixed, preserving the reliability of smart home systems. Similarly, Zamani et al. present a deep-learning approach using autoencoders with 1D-CNN and TCN backbones for time-series anomaly detection in smart homes. Their method effectively differentiates between normal and abnormal energy consumption patterns, particularly in dishwashers and refrigeration units, enhancing the ability to identify and address unusual consumption behaviors [26].

2.3. Contribution of This Paper

Despite the existing research on simulating energy consumption and the comparatively underexamined topic of simulating home appliance errors, there is still a significant gap in developing models that can simulate errors in home appliances with heterogeneous patterns. This paper addresses this gap by presenting a novel approach for simulating errors in home appliance power consumption patterns. Unlike existing studies, which primarily concentrate on modeling normal consumption patterns or detecting anomalies using supervised algorithms, our research introduces a novel unsupervised method capable of producing a wide range of error patterns in home appliances. Specifically, this method allows for the simulation of errors without requiring ground-truth or predefined anomaly data, making it highly applicable to real-world and real-time scenarios. As a result, the simulated anomaly patterns may be used for developing robust anomaly detection systems and predictive maintenance strategies in smart homes, where the diversity of error patterns can vary significantly based on usage habits and environmental conditions. By providing diverse error patterns, our method enhances the reliability and efficiency of smart home systems. This innovative contribution marks a substantial advancement in the field of anomaly detection, offering a valuable resource for mitigating appliance malfunctions in real-world scenarios.

3. Methodology

In this section, different types of home appliance anomalies are presented. Additionally, an extensive analysis of the datasets used, including types of home appliances and data preprocessing steps, is illustrated. Finally, the algorithms and techniques used to simulate error patterns in the power consumption data are presented.

3.1. Overview of Anomalies in Home Appliance Power Consumption Patterns

Anomalies in home appliance power consumption patterns refer to deviations from normal operational consumption patterns. These deviations can indicate various issues, from minor inefficiencies to major malfunctions [6]. Traditional methods for detecting anomalies often rely on predefined thresholds or supervised learning algorithms that require labeled datasets that indicate normal and faulty states [27]. However, these approaches have limitations and also require extensive labeled data [9]. Moreover, they have difficulty generalizing across different types of appliances and usage patterns [28]. Moreover, typically types of anomalies are referred to as:

i

Point Anomalies, as Figure 1 shows, refers to single data points that deviate significantly from the rest of the data (e.g., a sudden spike in power consumption) [29].

However, this spike may be misleading as it could also be a normal pattern such as a fridge compressor starting or a sensor reading error (Figure 2) [30]. When a fridge compressor kicks in, it can cause a temporary surge in power usage that may appear as an anomaly but is actually part of normal operation [31]. Additionally, such spikes can also be caused by sensor reading errors, leading to false positives in anomaly detection [32]. Consequently, while point anomalies are widely used, they should be approached with caution or carefully calibrated to avoid false positives.

ii

Contextual Anomalies (Figure 3) refers to data points that are considered anomalies within a specific context (e.g., high power consumption during an idle phase) [33].

High power consumption during an idle phase of a dishwasher might be flagged as an anomaly. However, this could be misleading as it might reflect legitimate operational variations, such as the dishwasher entering a self-cleaning cycle or handling a particularly heavy load [34]. Additionally, sensor errors (Figure 4) or variations in water pressure can also cause temporary spikes in power consumption that appear anomalous but are actually within normal operating parameters [35], while contextual anomalies are useful, they should be carefully analyzed and calibrated to avoid false positives.

Due to two main reasons, namely, the lack of labeled anomaly data to run supervised anomaly detection and the false positives arising from detecting anomaly patterns (whether point or contextual), a different method is employed in this paper. Traditional anomaly detection often relies on labeled datasets, which are challenging to obtain and can lead to inaccurate results due to false positives from normal operational patterns. In this paper, we use an unsupervised method that does not rely on detecting individual anomaly points.

Instead, after carefully examining patterns in home appliances, the method is based on the increase in power consumption caused by malfunctions, which translates to the prolonged duration of operational cycles [3,4]. Minor or major malfunctions typically result in increased power consumption, which is detected in the home appliance usage pattern as a prolonged operational cycle (time duration) compared to a normal cycle. For instance, a malfunctioning dishwasher may take significantly longer to complete a cycle due to issues like clogged filters or faulty sensors, leading to extended power usage beyond normal operational parameters. By focusing on these prolonged consumption patterns rather than isolated spikes or contextual anomalies, our method provides a more reliable and practical solution for real-time anomaly detection in smart home environments.

In developing our simulation model, various potential sources of error, including the assumptions underlying the model, the impact of missing data, and the effects of different types of anomalies on the simulation’s accuracy are considered. One key factor is that in appliance malfunctions leading to prolonged operational cycles, the extension to the operational cycle needs to be sufficiently long to be detected; if the prolonged cycle is less than one minute longer, this may be lost in the data. Additionally, the presence of missing data can significantly affect the model’s performance. Our analysis indicates that if more than 30% of the data are missing, the accuracy of the simulation decreases substantially. Otherwise, the data are cured to maintain its integrity. It is important to note that errors in the simulation model will not occur due to the robust design of our methodology, which incorporates real-time data adaptability.

As a result, in this paper, the total simulated anomaly patterns refer to a comprehensive set of prolonged operational cycles that reflect various types of malfunctions in home appliances as depicted in Figure 5. An anomaly is determined to be present in each of the patterns due to the existence of a prolonged operational cycle. These patterns are generated without relying on predefined labels or ground truth, making them applicable for developing robust anomaly detection systems. By simulating the extended duration of operational cycles indicative of different malfunction scenarios, this approach creates realistic anomalies while enhancing the reliability and efficiency of smart home systems. By focusing on prolonged consumption patterns rather than isolated spikes or contextual anomalies, the suggested method provides a more reliable and practical solution for real-time anomaly detection in smart home environments.

3.2. Types of Anomalies by Severity

The devices that are either widely owned (e.g., fridges [36]) or have high energy consumption (e.g., water heaters [36]), making them crucial for mitigating energy consumption due to overconsumption and errors, were selected for the suggested methodology. There is an additional important category not studied here: appliances focused on space heating and cooling. These were not included in the analysis due to the variety of systems available, such as air conditioners (AC), HVAC systems, heat pumps, electric radiators, gas heaters, and more. The lack of public data for all these systems, and the challenge of accurately measuring their consumption in integrated central systems, further complicated their inclusion. In future work, there is a plan to extend our analysis to include these heating and cooling appliances to further enhance our understanding of energy consumption and anomaly detection in these systems.

Specifically, according to the U.S. Department of Energy, water heating typically accounts for 13% to 17% of a household’s energy usage, making it one of the most significant energy expenses after heating and cooling [37]. Fridges and freezers, which are in almost every residence [38], also contribute to substantial energy use. As a result, the common or wasteful home appliances selected for analysis include fridges and freezers, washing machines, dishwashers, dryers, and water heaters. These appliances were selected as they are prevalent in most households while having significant energy demands. Fridges and freezers operate continuously, making them major contributors to household energy consumption [39]. Moreover, washing machines and dishwashers are used frequently and involve a water-heating phase, which increases the energy they consume [40]. Dryers are also known for their high power consumption, especially during drying cycles [41]. Finally, water heaters also represent substantial energy use in homes [42]. Therefore, these are the devices whose errors should be simulated to understand and mitigate their impact on overall energy consumption.

All home appliances may experience minor or major errors/malfunctions while continuing to operate, with these errors often remaining unnoticed by the homeowner. They continue to operate, but the consumption of the home appliance increases as the operation cycles are prolonged [3,4,43]. The duration of the prolongation depends on the type of malfunction. Minor errors are typically small malfunctions that do not immediately affect the appliance’s ability to perform its primary function but can cause it to operate less efficiently. Examples include slightly worn door seals on fridges or partially clogged lint filters in dryers. These minor issues lead to increased energy consumption but the appliance still appears to operate normally [3]. Conversely, major errors significantly impact an appliance’s performance and will eventually lead to complete failure [4]. These errors often remain unnoticed until the home appliance breaks. Examples include a faulty compressor in a fridge or a broken heating element in a water heater. These errors not only increase energy consumption substantially but also cause the appliance to underperform or stop working altogether.

Figure 6 provides an overview of the percentage of minor and major errors for each home appliance [43], as well as the percentage increase in energy consumption associated with specific malfunctions [43]. For instance, faulty thermostats in fridges can increase energy use by 23% [3,44]. Similarly, water heater major malfunctions can raise energy consumption by up to 35% [45]. Dishwashers with heating element issues can see a 15% increase in energy usage [46]. Additionally, inefficient heating elements in dryers can lead to a 25% increase in energy consumption [47].

3.3. Home Appliance Consumption Pattern Analysis

Understanding the power consumption patterns of home appliances is crucial for creating accurate simulated anomaly patterns. Each appliance has unique consumption characteristics influenced by its function, usage frequency, and operational cycles. This section provides a detailed analysis of the power consumption patterns for five common home appliances: fridge, washing machine, dishwasher, dryer, and water heater. This section also addresses the importance of granularity in real-time applications, emphasizing the need for high-resolution data to accurately detect anomalies and optimize appliance recognition.

3.3.1. Granularity and Its Importance in Real-Time Applications

When considering methods, especially for real-time and real-world applications, granularity is critical, particularly in energy consumption analysis. High-time resolution data are essential because they preserve the detailed patterns necessary for accurate analysis. Specifically, for methods such as Non-Intrusive Load Monitoring (NILM), a resolution smaller than 1 min, typically around 1 s, is often required [48]. These high-resolution data allow for the detection of subtle and rapid changes in power consumption that are crucial for recognizing anomalies and appliance-specific usage. However, for tasks such as load prediction, PV prediction, anomaly detection, and predictive maintenance in energy consumption, a 1 min resolution has been found to offer a good balance [49,50,51]. This maintains sufficient detail to detect anomalies and predict load changes accurately while managing data volume effectively.

Overall, in real-time applications, a balance must be struck between data management and pattern efficacy. For example, storing 1 s resolution data results in 86,400 data points per day, equating to approximately 31.5 million data points per year. In contrast, 1 min resolution data results in 1440 data points per day, or around 525,600 data points per year. This significant reduction in data volume (approximately 98.3% less) is crucial for practical data storage and processing in real-world applications. For long-term applications, such as smart home energy management, the implications of data storage are even more pronounced. Continuously storing 1 s resolution data can quickly lead to memory exhaustion and increased costs for data storage infrastructure. This can be particularly wasteful for applications that need to operate over many years. In contrast, 1 min resolution data provide a more manageable data volume that balances the need for detailed consumption patterns with the practicalities of long-term data storage.

As a result, the proposed solution is tested and recommended for use with a 1 min resolution as this provides the perfect balance between data richness and data management, while it can be adapted for resolutions of less than 1 min, its efficacy diminishes significantly when applied to resolutions greater than 5 min. Beyond this threshold, prolonged patterns might be misclassified as normal patterns, reducing the accuracy of anomaly detection and appliance recognition. Due to the prolonged patterns being subsumed within normal patterns, the anomaly patterns will become unrecognizable. Therefore, a 1 min resolution is optimal for maintaining a detailed yet manageable dataset, ensuring effective anomaly detection and appliance recognition.

3.3.2. Fridge

Fridges show a relatively stable power consumption pattern with periodic peaks. The power consumption pattern is composed of two phases: the cooling phase and the idle phase. During the active cooling phase, the compressor runs to cool the interior of the fridge. The power consumption in this phase is at its highest level as both the compressor motor and fans are operating. The duration of this phase depends on the efficiency of the fridge, as well as the internal and external temperatures, and may last from a few minutes to hours.

In the idle phase, the power consumption drops significantly, as only the sensors, lights, and display panels are active. The compressor is turned off when the set temperature has been achieved. The fridge remains in this phase until the temperature inside exceeds a certain set point. Figure 7 presents the power consumption pattern of a fridge.

3.3.3. Washing Machine

Washing machines are a commonly used home appliance that greatly contributes to the convenience of daily life. These devices come in various types, such as top-loading and front-loading machines, each with its unique operational characteristics. In this work, data from front-loading washing machines was utilized.

All types of washing machines exhibit cyclical power consumption patterns characterized by distinct phases: heating, washing, and spinning. The heating phase is the most power-consuming, with power consumption peaks due to the activation of the heating elements. This phase is important for raising the water temperature to the desired level, which is necessary for the selected washing program. There is another phase before the heating phase called the pre-washing phase, but this appears only in specific washing programs with a pre-washing process.

Figure 8 presents the pattern of a washing cycle without a pre-washing phase. As can be observed, the heating phase is the most power-consuming, accounting for almost 90% of the total power consumption. The washing and spinning phases have lower power consumption, and the duration of these phases depends on the operation program selected by the user.

3.3.4. Dishwasher

Dishwashers have a multi-phase power consumption pattern, similar to washing machines. The phases consist of the pre-wash, wash, rinse, and dry phases. The different phases are presented in Figure 9.

The initial phase is the pre-washing phase. During this phase, a brief period of power consumption occurs as the machine prepares for the main washing phase. This involves pumping in water and slightly heating it. The next phase is the washing phase, in which a significant increase in power consumption is observed. The heating element works to maintain the water temperature at an optimal level, while the motor drives the spray arms to distribute water and detergent. The power peaks in this phase can be substantial, reflecting the combined energy demands of heating and mechanical action.

The rinsing phase includes the removal of water from the machine. The power consumption decreases compared to the washing phase. The motor continues to operate, and in some models, additional heating elements are activated to ensure the water is at the right temperature for effective rinsing. The last phase includes the drying process involving passive air drying or active heat drying. In models with heat drying, the power consumption peaks again as the heating element works to dry the dishes. This phase can show varied power usage depending on the drying method and settings used.

3.3.5. Dryer

Dryer power consumption patterns depend on the selected program and the main feature of all programs is the high power consumption due to the heating elements. The power usage is relatively consistent throughout the drying cycle, with minor variations depending on the cycle’s intensity and the load size. The primary phases are the warming-up, heating, and cooling-down phases and are presented in Figure 10.

In the warm-up phase, the dryer begins to heat up, and power consumption rises as the heating elements engage. This phase is relatively short. During the heating phase, the dryer operates at a high power level. The heating elements operate continuously or intermittently to maintain a high temperature, while the motor keeps the drum rotating. Despite minor fluctuations based on the drying cycle’s intensity, power consumption stays high and relatively stable. The cool-down phase is the final stage of the dryer power consumption pattern. In this phase, the dryer enters a cool-down period where the heating elements turn off, and the drum continues to rotate to cool the clothes. Power consumption drops significantly during this phase.

3.3.6. Water Heater

Water heater power consumption typically exhibits a relatively constant pattern, with high power usage during heating periods and negligible consumption during idle periods. The thermostat settings and hot water usage frequency influence the consumption pattern.

The power consumption pattern of a water heater is shown in Figure 11. It is constantly high during active heating, which indicates the energy needed to heat the water. The consumption reduces significantly once the target temperature is attained and stays low during standby periods. In some models, occasional power spikes may occur due to reheating cycles to maintain the water temperature.

The power consumption patterns of the aforementioned home appliances have distinctive characteristics that are important for developing simulated anomaly situations. The fridge shows stable patterns with periodic peaks, the washing machine displays cyclical patterns with three main phases, the dishwasher displays multiple peaks corresponding to their operational phases, the dryer has relatively consistent high consumption, and the water heater shows high consumption during heating periods.

Table 1. Statistical Features of Power Consumption for Various Home Appliances.

Appliance	Mean Power Consumption	Peak Power Consumption	Cycle Duration (Avg)	Frequency of Peaks	Power Consumption Variance
Fridge	Low	Moderate (Compressor, Defrost)	Continuous	Moderate (Periodic Peaks)	Low
Washing Machine	Moderate	High (Heating, Spinning)	1–3 h	High (Cyclical Peaks)	High
Dishwasher	Moderate	High (Wash, Dry Phases)	1–2 h	High (Multiple Phases)	High
Dryer	High	High	1–1.5 h	Low (Consistent)	Moderate
Water Heater	Low (Idle)/ High (Heating)	High	Varies	Low (Heating Spikes)	High

presents the values for six statistical features. For the mean value, “Low” represents a value lower than 100 watts, “Moderate” shows a value between 100 and 500 watts, and “High” represents values above 500 watts. For peak power consumption values, “Moderate” includes values between 500 and 1500 watts, and “High” represents values above 1500 watts.

For cycle duration, “Continuous” means that the appliance operates consistently without defined cycles, while “Varies” indicates that the duration of the appliance’s operating cycle can change significantly based on usage patterns and settings. The frequency of peaks characterizes whether a consumption pattern is consistent or periodic. “Low” indicates stable power usage, “Moderate” reflects periodic increases in power consumption, and “High” indicates regular significant power usage fluctuations.

Finally, the values of power consumption variance indicate that “Low” corresponds to a relatively stable pattern with minimal fluctuations, “Moderate” reflects variations in power consumption due to changes in operating states, and “High” reflects significant fluctuations in power consumption, often due to different operational phases or cycles. This detailed analysis serves as a foundation for generating simulated data that can be used to train and validate advanced anomaly detection algorithms.

3.4. Simulation Techniques

In this section, the simulation techniques used to generate the simulated anomaly datasets are presented. These techniques involve algorithms designed to identify sequences of normal power consumption, introduce simulated errors, and create a comprehensive dataset of power consumption anomaly patterns.

Different workloads can influence the operational efficiency and power consumption patterns of appliances, such as washing machines and dishwashers, leading to different power consumption patterns as well as anomaly patterns. Varying usage scenarios, including differences in load sizes, fabric types, and settings, result in changes in energy consumption and operational behavior. To address these variations, different workload scenarios in the analysis and simulation algorithms should be used.

3.4.1. Methodology Overview and Application to Diverse Appliance Types

The proposed methodology for simulating anomaly patterns in home appliance power consumption involves several key steps, as depicted in Figure 12, which provides a comprehensive flow chart of the process. This methodology is designed to be applicable across a wide range of home appliances, each with unique power consumption characteristics.

Select Device and Study the Power Consumption Pattern: The process begins with selecting a specific device, followed by a detailed study of its power consumption pattern. This step is crucial for understanding the normal operational behavior of the appliance, which varies significantly between different types of devices. For example, refrigerators display consistent patterns with regular cycles of high power consumption when the compressor is running, while washing machines exhibit distinct peaks during specific intervals of a washing cycle. This stage ensures that the model is not limited to a specific set of home appliances, allowing the approach to be generalized to other types of devices, ensuring that the model has sufficient scope.

Data Collection and Preprocessing: A key aspect of the methodology is the collection and preprocessing of time-series power consumption data. This step involves cleaning the data to remove outliers, filling in missing values, and ensuring consistency in data granularity, typically set at a one-minute interval. This preprocessing step is essential for managing the data’s variability and noise, which are common in real-world environments. Importantly, this process is conducted offline, meaning that the data preprocessing is performed locally rather than in real-time. This approach allows for more thorough and robust data handling, free from the constraints of real-time processing. Consequently, the preprocessed data, once cleaned and structured, can then be utilized effectively in real-time applications. By implementing these robust preprocessing techniques, the model’s ability to handle diverse and noisy data is significantly enhanced, making it well-suited for practical, real-time applications.

Detection of Operational Power Consumption Pattern: After preprocessing the data, the next step is to detect the operational power consumption pattern of each appliance. This involves analyzing the power consumption data and dividing the data into distinct sub-sequences that correspond to the various operational patterns of the device. For example, refrigerators typically exhibit a consistent daily pattern due to their regular cooling cycles, while washing machines show power consumption only during specific intervals when a washing program is running.

Extraction of Normal Operational Phases: After identifying the operational cycles, the normal operational phases are extracted. This step focuses on segmenting the power consumption data into distinct sub-sequences that represent the different phases of an appliance’s operation, such as the heating, washing, and drying phases of a washing machine.

Definition of Malfunction Types: In this step, different types of malfunctions are defined based on their impact on the appliance’s operational phases. Malfunctions are categorized into minor and major types, with each category reflecting a different level of deviation from normal operation. This categorization is crucial for the subsequent simulation of anomaly patterns and allows the model to be applied without the use of labeled data. The specific percentages of energy increase associated with minor and major malfunctions, as well as the percentage increase related to specific types of malfunctions, are derived from the study of the power consumption data in the previous step. This data-driven approach ensures that the model accurately reflects the real-world impact of different malfunctions on energy consumption, thereby enhancing the reliability and applicability of the simulated anomaly patterns.

Creation of Anomaly Patterns: Using the defined malfunction types, anomaly patterns are created by altering the duration and intensity of specific operational phases. For instance, in a washing machine, the heating phase might be extended to simulate a faulty heating element. These simulated anomalies are integrated into the normal consumption patterns to produce a dataset that reflects both normal and faulty operation scenarios.

Export of Anomaly Power Consumption Patterns: The final step involves exporting the simulated anomaly patterns, which can be used for training and testing anomaly detection models.

Overall, This approach addresses the challenge of the limited availability of labeled data by generating a comprehensive set of simulated data that can be used to improve the accuracy and robustness of anomaly detection systems. The proposed methodology can be used for a wide range of home appliances. In this paper, five of the most commonly used home appliances were used to create simulated anomaly datasets.

3.4.2. Fridge

The procedure for simulating anomaly patterns in fridges involves two steps. The first step is identifying each sub-sequence within the total power consumption. A sub-sequence is defined as the duration of power consumption from the moment the compressor is enabled until it is disabled. The detailed procedure is provided in Algorithm 1. The algorithm takes as input the power consumption data organized in a time series. A threshold T is then defined to detect when the compressor is off. This threshold is crucial for accurately identifying the compressor’s operational cycles and it was empirically set at 20 watts. This value was chosen based on typical power consumption patterns of fridges, where the power consumption drops significantly below 20 watts during idle phases when the compressor is off. This threshold ensures that only periods of active compressor operation are selected.

Algorithm 1 Detect Sub-Sequences

1: Input: Power consumption data y 2: Output: List of sub-sequences with power consumption, last positions 3: Define threshold T 4: Identify indices where power consumption y is below threshold T for k consecutive points $$indices=\left\{i\right|\forall j\in [i,i+k-1],y_j<T\}$$ 5: Split data into sub-sequences based on identified indices $$sub\_sequences=\left\{\mathrm{data}\right[i:j\left]\right|i,j\in \mathrm{indices}\}$$ 6: Select sub-sequences longer than 5 min $$selected\_sequences=\{s|s\in \mathrm{sub}\_\mathrm{sequences}\mathrm{and}\mathrm{length}(s)>5\mathrm{minutes}\}$$ 7: Return: List of sub-sequences with power consumption, last positions

In the next step, the indices in which the power consumption is below T for k consecutive time intervals are identified. The value of k was defined as 2 min. This means that if the power consumption is below the threshold T for more than 2 min, this indicates that the compressor is off, as there is no power consumption attributed to the compressor’s operation. The following equation describes the procedure:

$$\mathrm{indices}=\left\{i\right|\forall j\in [i,i+k-1],y_j<T\},$$

(1)

where y is the power consumption data, i is the starting index of the sub-sequence, j is the index within the range of consecutive points, and k is the number of consecutive points below the threshold T.

Subsequently, the power consumption data are split into sub-sequences based on these indices, and only sub-sequences with durations exceeding 5 min are selected. The minimum size of the sub-sequences was set to 5 min. According to [3], the operation cycle of the refrigerator is typically longer than 5 min. Therefore, sub-sequences shorter than 5 min are removed. The following equations describe the procedure:

$$\mathrm{sub}\_\mathrm{sequences}=\left\{\mathrm{data}\right[i:j\left]\right|i,j\in \mathrm{indices}\},$$

(2)

where $\mathrm{data}[i:j]$ represents a sub-sequence of power consumption data from index i to j.

$$\mathrm{selected}\_\mathrm{sequences}=\left\{s\right|s\in \mathrm{sub}\_\mathrm{sequences}\mathrm{and}\mathrm{length}\left(s\right)>5\mathrm{minutes}\},$$

(3)

where s represents a sub-sequence from the set of sub-sequences. This selection is based on the assumption that the compressor requires more than 5 min of operation to maintain the fridge’s temperature as mentioned before.

The outputs are a list of sequences representing continuous power consumption periods below the specified threshold T as well as the last position j of each sub-sequence relative to the input dataset.

The second step involves the simulation of anomaly patterns. In this step, power consumption anomaly patterns are generated. According to [3], most of the total power consumption is consumed when the compressor is enabled. Thus, anomaly patterns are generated during periods when the compressor is enabled. The complete procedure is described in Algorithm 2. The outputs from Algorithm 1 and the user-defined error percentage serve as inputs. For each final position value j, the size of the extension is calculated based on the size of the sub-sequences and the error percentage (e). To calculate the extension, the weighted mean of the last three instances is used, with weights proportional to their values. The following formula was used to calculate the extension values: Given the last n instances of power consumption values $\{x_{p-n},x_{p-(n-1)},\dots ,x_{p-2},x_{p-1}\}$, the weighted mean is calculated as:

$$\mathrm{weighted}\_\mathrm{mean}\_\mathrm{last}\_\mathrm{n}={\displaystyle \frac{{\sum }_{i=1}^{n}i·x_{p-n+i-1}}{{\sum }_{i=1}^{n}i}},$$

(4)

where n is the number of instances for calculating the weighted mean, $(x_{p-n},x_{p-(n-1)}$, $\dots ,x_{p-2},x_{p-1})$ is the power consumption values at the last n positions in the sequence, and $x_{p-k}$ represents the power consumption value at position $p-k$. In the final step, a random value is added on each $\mathrm{weighted}\_\mathrm{mean}\_\mathrm{last}\_\mathrm{n}$ and the output is the anomaly power consumption pattern for each sub-sequence.

Algorithm 2 Simulated Error

1: Input: Power consumption data y, last positions P, error percentage e 2: Output: Power consumption anomaly patterns 3: for each position $p\in P$ do 4:     Calculate the size of the extension E based on the size of the sequences and the error percentage e $$E=\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{sequence}\times e$$ 5:     Calculate the weighted mean of the last three instances with weights proportional to their values to create anomaly patterns $$weighted\_mean\_last\_n={\displaystyle \frac{{\sum }_{i=1}^{n}i·x_{p-n+i-1}}{{\sum }_{i=1}^{n}i}}$$ 6:     Add the anomaly patterns to the sequence at the specific position p $$y_p=\mathrm{weighted}\_\mathrm{mean}\_\mathrm{last}\_\mathrm{n}\times (1+\mathrm{random}.\mathrm{uniform}(0.001,0.009))$$ 7: end for 8: Return: Power consumption anomaly patterns for each sub-sequence

3.4.3. Washing Machine, Dishwasher, and Dryer

Algorithms 3 and 4 describe the procedure for creating anomaly patterns in the existing power consumption patterns of washing machines, dishwashers, and dryers. The procedure consists of two steps. The first step is responsible for extracting the heating phases of the consumption pattern, and the second step is responsible for extending the consumption pattern based on the category of malfunctions selected by the user.

Algorithm 3 Extract Heating Phases of a Washing Machine

1: Input: Power consumption data y over time x 2: Output: Segments with the heating phases 3: Fit a mathematical equation to the power consumption pattern: $f\left(x\right)$ 4: Calculate the first derivative to find the slopes: $f^{\prime }\left(x\right)={\displaystyle \frac{d}{dx}}f\left(x\right)$ 5: Set a threshold T to keep only the high changes in the pattern: $$High\_Changes=\{x_i|y_i>T\mathrm{and}|f^{\prime }\left(x_i\right)|>0.5\times max(|f^{\prime }\left(x\right)\left|\right)\}$$ 6: if$\mathrm{len}(\mathrm{High}\_\mathrm{Changes})mod2\ne 0$then 7:     Recalculate High Changes with reduced threshold: $$High\_Changes=\{x_i|y_i>T\mathrm{and}|f^{\prime }\left(x_i\right)|>0.2\times max(|f^{\prime }\left(x\right)\left|\right)\}$$ 8: end if 9: Filter out indices based on the final threshold 10: Return: Segments with the heating phases

Specifically, the input for Algorithm 3 is the power consumption data and the output includes segments with the detected heating phases. Initially, the algorithm fits a mathematical Equation ($f\left(x\right)$) to the power consumption pattern. The next step involves identifying the portion of the data that corresponds to the heating phase by calculating the first derivative of the fitted equation, $f^{\prime }\left(x\right)={\displaystyle \frac{d}{dx}}f\left(x\right)$. Examples of the calculation of the first derivative for a washing machine, dishwasher, and dryer are presented in Figure 13. As can be observed, calculating the first derivative makes it easy to detect when the heating phases start. The starting point of the heating phase is when a big change in the first derivative is observed.

Algorithm 4 Extend Heating Phase

1: Input: Heating phase segments, power consumption data, percentage anomaly 2: Output: Extended pattern 3: Find the end of the heating phase 4: Calculate the extension based on the percentage anomaly:     $extension=$ length of heating phase segments × percentage anomaly     5: Extend pattern with random adjustment: 6: for each value in the extended segment do     7:     $adjustment=\mathrm{random}.\mathrm{uniform}(0.001,0.009)$ 8:     $new\_value=$ last value of the heating phase $\times (1+adjustment)$     9: end for 10: Append the extended segment with the random adjustments to the original dataset

The results of the first derivative analysis are used in the next step to detect when the heating phase starts. A threshold T is set to identify significant changes in the power consumption pattern. For the washing machine and dishwasher, where the high values of power consumption are very close, the same threshold value of 10 was selected. This value was chosen based on typical power consumption patterns, in which significant power spikes occur during the heating phase. Conversely, for the dryer, where the changes in power consumption values are smaller, a threshold value of 2 was selected. The algorithm then identifies points where the power consumption $y_i$ exceeds the threshold T and where the absolute value of the first derivative $|f^{\prime }\left(x_i\right)|$ is greater than 50% of the maximum absolute value of the first derivative. The following equation describes the threshold:

$$\mathrm{High}\_\mathrm{Changes}=\{x_i|y_i>T\mathrm{and}|f^{\prime }\left(x_i\right)|>0.5\times max(|f^{\prime }\left(x\right)\left|\right)\},$$

(5)

where $y_i$ is the power consumption value at time $x_i$. If the number of identified high-change points is odd, the threshold is recalculated with a reduced value (20% of the maximum absolute value of the first derivative) to ensure an even number of high-change points. It is not possible to have an odd number of high-change points as the heating phase must have both a starting point (enable) and an ending point (disable). Finally, the algorithm filters out indices based on the recalibrated or final threshold to accurately identify the segments of the heating phases.

Algorithm 4 takes as input the segments with the heating phase from the previous algorithm, and the output is an extended consumption pattern that includes the anomaly pattern. The first step involves identifying the endpoint of the heating phase within the power consumption data. Based on the user’s input (percentage anomaly), the extension of the existing consumption pattern is then calculated. To determine the values for the new extended values (anomaly pattern), a small random adjustment is applied. Finally, the extended segment, with random adjustments, is appended to the original dataset to form the extended heating phase pattern.

3.4.4. Water Heater

The procedure for simulating anomaly patterns in the water heater focuses on detecting the heating phase, which can be identified using a pre-defined threshold. Unlike other appliances with multiple phases, the water heater primarily has a single heating phase in all patterns. For water heaters, the threshold T was set at 1000 watts. This threshold was chosen because water heaters typically consume a large amount of power during their heating phase, and setting a high threshold ensures that only significant power consumption periods are detected. The process starts by identifying the heating phase within the power consumption data. To identify the heating phase, Algorithm 1 is used.

As input data, the device’s power consumption is used, organized in a time series. A threshold T is defined to detect when the heater is in operation. The next step includes identifying indices where the power consumption exceeds the threshold T for consecutive time intervals of 2 min. This means that if the power consumption is above the threshold T for more than 2 min, this indicates that the resistance is on, and the water heater is enabled. Then, the power consumption data is split into sub-sequences based on the identified indices, and only those sub-sequences where the duration exceeds a minimum operational time of 3 min are selected. Only sub-sequences longer than 3 min are selected to ensure that short, insignificant spikes in power consumption are not considered. This duration was chosen based on the typical operational characteristics of water heaters and other home appliances.

The extracted sub-sequences are used as input in Algorithm 4. First, the algorithm identifies the end point of each heating phase within the power consumption data. Then, it calculates the extension size based on the length of the heating phase segments and the specified percentage anomaly selected by the user, determining how much longer the heating phase will be extended to simulate anomalies. Finally, the algorithm extends the pattern by applying a small random adjustment to each value in the extended segment, ensuring the anomalies appear realistic and append the anomaly patterns to the original dataset, forming an extended heating phase pattern that includes the simulated anomaly patterns.

3.5. Time Complexity of the Proposed Methodology

Time complexity is a measure of the amount of computational time that an algorithm requires as a function of the size of the input data [52]. It is a crucial metric for evaluating the efficiency of an algorithm, particularly for real-time applications where rapid processing and quick response times are essential. Estimating the time complexity helps in understanding how the performance of the algorithm scales with increasing data sizes, ensuring that it can handle large datasets efficiently without significant delays.

The time complexity of the proposed methodology can be estimated based on the main steps involved in the simulation of anomaly patterns. An outline of the steps and their respective time complexities is as follows:

I

Step 1. Data Preprocessing:

• Removing outliers involves scanning the dataset once, so this step has a time complexity of $O\left(n\right)$, where n is the number of data points.
• Filling missing values also requires scanning the dataset, hence $O\left(n\right)$.

II

Step 2. Identifying Sub-Sequences (Algorithm 1):

• This step involves scanning the power consumption data to identify points below a threshold, which is $O\left(n\right)$.
• Splitting data into sub-sequences and selecting those longer than 5 min also requires a single pass through the data, resulting in $O\left(n\right)$.

III

Step 3. Simulating Anomaly Patterns (Algorithm 2):

• For each sub-sequence identified, calculating the extension size and applying a weighted mean calculation involves constant time operations for each sub-sequence, leading to $O\left(k\right)$, where k is the number of sub-sequences.
• Since $k\le n$, this step can be approximated as $O\left(n\right)$.

IV

Step 4. Extracting Heating Phases (Algorithm 3):

• Fitting a mathematical equation to the power consumption pattern and calculating the first derivative are operations that depend on the complexity of the fitting algorithm. Assuming a linear or polynomial fitting, this can be carried out in $O\left(n\right)$.
• Identifying high-change points and recalculating thresholds involves scanning the data and applying conditions, resulting in $O\left(n\right)$.

V

Step 5. Extending Heating Phase (Algorithm 4):

• extending the pattern and applying random adjustments involve a single pass through the segments, giving a complexity of $O\left(m\right)$, where m is the length of the heating phase segments.
• Since, $m\le n$, this step is also $O\left(n\right)$.

Combining all these steps, the overall time complexity of the proposed methodology can be estimated as $O\left(n\right)$, where n is the number of data points in the power consumption dataset. This linear time complexity indicates that the suggested method is efficient and scalable for large datasets. Furthermore implementing optimizations, like parallel processing and efficient data structures, will further enhance performance, ensuring the suggested approach can handle scenarios involving large datasets effectively.

4. Simulation Results and Analysis

In this section, the datasets used for the simulation of anomaly patterns are presented. Specifically, for each device in the experiment set-up, the normal consumption pattern of the device is compared to the simulated anomaly pattern to illustrate the results clearly. In addition, results on the simulation data for each device, along with a statistical analysis, are presented in this section. The analysis includes examining the distribution, frequency, and impact of the simulated anomalies on the overall power consumption patterns. This comprehensive evaluation helps in understanding the effectiveness of the anomaly simulation techniques and their applicability in real-world scenarios.

4.1. Dataset

The datasets used for the simulations were collected from various household appliances, namely, fridges, washing machines, dishwashers, dryers, and water heaters. For each home appliance, different setups and workloads were used.

Table 2. Experimental setup and selected parameters for each appliance.

Appliance	Load Size	Program Settings	Conditions
Washing Machine	Bellow half-load	Normal	Water Temperature: 40 °C
	Half-load	Normal	Water Temperature: 40 °C
	Full-load	Normal	Water Temperature: 40 °C
Dishwasher	Half-load	Normal	Water Temperature: 50 °C
	Full load	Normal	Water Temperature: 50 °C
Fridge	N/A	N/A	Door Opening: Low Ambient Temp: 18–28 °C Internal Temp: 2–4°C
Dryer	Bellow half-load	Normal	Temperature: 60 °C
	Half-load	Normal	Temperature: 60 °C
	Full load	Normal	Temperature: 60 °C
Water Heater	N/A	N/A	Usage Frequency: Low Thermostat: 40 °C
	N/A	N/A	Usage Frequency: Medium Thermostat: 50 °C
	N/A	N/A	Usage Frequency: High Thermostat: 50 °C

presents the different parameters used. For the washing machines, a normal washing cycle was selected with three different loads (below half-load, half-load, and full-load). For the dishwasher, the normal program with a water temperature of 50 °C was selected with two load sizes (half-load and full-load). For the fridge, a scenario with an ambient temperature between 18 and 28 °C and an internal temperature between 2 and 4 °C was used. For the dryer, the normal program was selected with a temperature of 60 °C and three different load sizes (below half-load, half-load, and full-load). Finally, for the water heater, three scenarios were selected with three usage frequencies and two different selected temperatures (40 °C and 50 °C).

Each dataset consists of time-series power consumption data recorded at one-minute intervals. Z-Wave smart plugs (FIBARO Wall Plug (FGWPF-102 ZW5)) were used to collect the power consumption data for each device. Data from nine different houses were collected. The duration of the collected data was 1.5 years and the total number of collected data points for an operational cycle of each fridge, washing machine, dishwasher, and water heater was 43.200, while for each dryer, it was 28.800.

Table 3. Overview of the experimental datasets.

Pilots	Fridge	Washing Machine	Dishwasher	Dryer	Water Heater
House 1	✓	✓	✓		✓
House 2	✓	✓			✓
House 3	✓		✓	✓
House 4	✓	✓	✓		✓
House 5	✓	✓		✓
House 6	✓	✓		✓	✓
House 7	✓	✓
House 8	✓		✓		✓
House 9	✓	✓
Total different Simulated Anomaly Patterns	8	7	8	7	6

presents an overview of the experimental datasets. The last row of the table describes the total number of the simulated anomaly patterns derived from Figure 6. Moreover, the datasets were preprocessed to remove outliers, fill missing values, and ensure consistency in data granularity.

4.2. Fridge

Figure 14 illustrates the power consumption of normal and abnormal operations with minor and major errors for the fridge. Minor errors, such as a partially malfunctioning compressor, result in a slight increase in power consumption, with anomalies contributing to a 7.5% increase in mean power consumption. In contrast, major errors, such as a faulty thermostat, lead to a significant increase in power consumption, with anomalies causing a 15.7% rise. The simulation was made for the duration of the cycle, ensuring that the anomalies are realistically integrated into the normal consumption patterns.

Table 4. Overview of the experimental datasets for the fridge.

Statistic	Normal Data	Anomaly Data (Minor)	Anomaly Data (Major)
Mean (W)	38.815	39.765	40.707
Median (W)	0	0	0
Standard Deviation (W)	59.472	58.668	57.881
Variance (W2)	3537.032	3442.019	3350.252
Skewness	2.860	2.832	2.804
Kurtosis	11.012	11.105	11.193

presents an overview of the data for the fridge. As observed, the mean power consumption during minor error scenarios increased from 38.815 W (normal) to 39.765 W, indicating that minor malfunctions cause a noticeable but relatively small increase in energy usage. The standard deviation and variance in power consumption slightly decreased, indicating that minor errors introduce consistent, low-level increases in energy consumption without causing large fluctuations. Finally, the skewness and kurtosis remained relatively stable, indicating that the distribution of power consumption values did not change significantly, even with the presence of minor errors. In the case of major errors, the mean power consumption increased to 40.707 W, the standard deviation and variance decreased further compared to minor errors, and the skewness and kurtosis remained almost at the same values with minor changes.

To ensure the reliability of the model and the collected experimental data, a series of statistical tests were conducted, including Levene’s test for equality of variances and the Shapiro–Wilk test to assess whether the power consumption data follows a normal distribution under different conditions. For Levene’s test, comparisons between the variances under normal and minor error conditions, as well as between minor and major error conditions were made. For the comparison between normal and minor error conditions, the null hypothesis ($H_0$) posits that the variances are equal, while the alternative hypothesis ($H_a$) suggests that the variances are different. Similarly, the equality of variances between minor and major error conditions was evaluated. The results of Levene’s test indicated that there were no significant differences in variances (p > 0.05) across these conditions (normal and minor errors $F_{static}$ = 1.128, p = 0.288, normal and major errors $F_{static}$ = 0.0662, p = 0.912), supporting the assumption of homogeneity of variances.

For the Shapiro–Wilk test, the null hypothesis ($H_0$) posits that the data are normally distributed. The tests were conducted for normal operation and minor error, as well as major error conditions. The results of the Shapiro–Wilk test for normal operation ($F_{static}$ = 0.999, p = 0.771), minor error ($F_{static}$ = 0.998, p = 0.608), and major error ($F_{static}$ = 0.998, p = 0.612) conditions indicated that the data follow a normal distribution (p > 0.05), justifying the normality of the data distributions.

4.3. Washing Machine

Figure 15 presents the power consumption of a washing machine during normal operation and under conditions with minor errors (10%) and major errors (30%). It can be seen that errors primarily affect the heating phase, which represents nearly 90% of the total power consumption. Minor errors cause a slight increase in power consumption during this phase, while major errors result in a more significant increase.

Table 5. Overview of the experimental dataset for a washing machine.

Statistic	Normal Data	Anomaly Data (Minor)	Anomaly Data (Major)
Mean (W)	45.805	49.649	59.940
Median (W)	2	2	2
Standard Deviation (W)	271.440	283.963	315.162
Variance (W2)	73,680.04	80,635.217	99,327.303
Skewness	6.936	6.583	5.842
Kurtosis	46.9309	42.029	32.599

summarizes the statistical features from data derived from a washing machine. As can be observed, the average power consumption while a minor error occurs increased from 45.805 W to 49.6499 W, indicating that even minor malfunctions result in a considerable rise in energy consumption. The standard deviation and variance in power consumption slightly increased, which means that the presence of minor errors causes some form of variability in energy consumption. The Skewness and kurtosis values decreased slightly, indicating a minor shift in the distribution of power consumption values. For major errors, the mean power consumption increased to 59.940 W, and both the standard deviation and variance increased significantly compared to minor errors. The Skewness and kurtosis values further decreased, reflecting a more substantial shift in the distribution due to major errors.

To ensure the reliability of the washing machine data, Levene’s test for equality of variances, and the Shapiro–Wilk test for normality were conducted. For Levene’s test of the comparison between normal and minor errors, the results were $F_{static}$ = 1.077 and p = 0.299, suggesting no significant difference in variances. However, for the comparison between normal and major error, the test yielded $F_{static}$ = 27.75 and p = $1.47$∗ $e^{-7}$ indicating a significant difference in variances (p < 0.05). When Levene’s test indicates a significant difference in variances, this suggests that the variability in power consumption is not consistent across different error conditions. This can be due to the different nature and impact of errors on the appliance’s operation.

For the Shapiro–Wilk tests, the results for normal operation ($F_{static}$ = 0.999, p = 0.761), minor errors ($F_{static}$ = 0.991, p = 0.612), and major errors ($F_{static}$ = 0.989, p = 0.601) indicated that the data is normally distributed as p > 0.05 in all cases.

4.4. Dishwasher

Figure 16 presents the power consumption of a dishwasher with minor (7.5%) and major (35%) errors. Similar to the washing machine, the power consumption increases noticeably with the presence of errors. Minor errors result in a modest increase in power usage during different phases, whereas major errors lead to a higher rise in overall consumption.

Table 6. Overview of the experimental dataset for a dishwasher.

Statistic	Normal Data	Anomaly Data (Minor)	Anomaly Data (Major)
Mean (W)	43.196	46.13	76.616
Median (W)	0	0	0
Standard Deviation (W)	287.393	297.860	388.938
Variance (W2)	82,594.825	88,721.146	151,273.000
Skewness	7.205	6.936	5.175
Kurtosis	50.191	46.348	24.874

provides a comprehensive summary of the statistics related to the dishwasher. When minor errors occur, the mean power consumption rises from 43.196 W (normal) to 46.13 W, indicating a moderate increase in energy consumption. This suggests that minor malfunctions result in a noticeable yet relatively small escalation in power usage. Similarly, the standard deviation and variance in power consumption also increase, as some variability is added to the energy consumption pattern with minor errors. A decrease in skewness and kurtosis values indicates a slight alteration in the distribution of power consumption values, indicating a minor shift in the data distribution due to these errors. Conversely, for major errors, a significant increase in the mean is observed, as well as in the standard deviation and variance, indicating a considerable rise in consumption variability. The skewness and kurtosis values decrease further, illustrating a more pronounced change in the distribution of power consumption values due to major errors. These changes underscore the critical effect of major errors, leading to higher energy consumption and greater fluctuations in power usage.

Levene’s test for equality of variances and the Shapiro–Wilk test for normality were conducted for the dishwasher normal, minor error, and major error dataset. For Levene’s test and the comparison between normal and minor errors, the results were $F_{static}$ = 1.605 and p = 0.205, suggesting no significant difference in variances. However, for the comparison between normal and major error, the test yielded $F_{static}$ = 110.766 and p = $1.91$∗$e^{-10}$ indicating a significant difference in variances (p < 0.05). Similar to the previous dataset of a washing machine, the results of Levene’s test suggest that the variability in power consumption is not consistent across different error conditions.

For the Shapiro–Wilk tests, the results for normal operation ($F_{static}$ = 0.999, p = 0.771), minor error ($F_{static}$ = 0.998, p = 0.608), and major errors ($F_{static}$ = 0.998, p = 0.611) indicated that the data are normally distributed as p > 0.05 in all cases.

4.5. Dryer

The real and simulated data for minor (7.5%) and major (35%) are presented in Figure 17. The error in this device is detected during the heating phase, which is almost 100% of the total power consumption of a dryer cycle.

Table 7. Overview of the experimental dataset for a dryer.

Statistic	Normal Data	Anomaly Data (Minor)	Anomaly Data (Major)
Mean (W)	71.949	76.910	94.032
Median (W)	0	0	0
Standard Deviation (W)	181.005	186.521	203.193
Variance (W2)	32,763.064	34,790.350	41,287.73
Skewness	2.172	2.062	1.736
Kurtosis	2.803	2.329	1.065

presents an overview of the extracted statistical features of an operated cycle of a dryer. For minor errors, the mean power consumption increases from 71.949 W (normal) to 76.910 W, suggesting that minor malfunctions cause a noticeable but modest rise in energy usage. In addition, the standard deviation and variance also increase, indicating variability in the power consumption when minor errors occur. The skewness and kurtosis values decrease slightly, suggesting a minor shift in the distribution of power consumption values. For major errors, a higher increase in power consumption is observed, highlighting the severe impact of these errors on energy usage. The standard deviation and variance show considerable increases, reflecting a greater variability in power consumption. Finally, the skewness and kurtosis values further decrease, indicating a more substantial shift in the distribution.

As with the other home appliances, to ensure the reliability of the analysis, Levene’s test and the Shapiro–Wilk test were applied. Levene’s test indicated a significant difference in variances between normal and major error conditions ($F_{static}$ = 16.702 and p = $4.492$ ∗ $e^{-5}$). This suggests that the variability in power consumption differs significantly between these conditions. Conversely, for the comparison between normal and minor errors, Levene’s test indicated $F_{static}$ = 1.1315 and p = 0.287, suggesting no significant difference in variances.

The Shapiro–Wilk test results for normal operation ($F_{static}$ = 0.994 p = 0.698), minor error ($F_{static}$ = 0.981, p = 0.178), and major error (W = 0.994, p = 0.612) conditions showed that the data follow a normal distribution (p > 0.05).

4.6. Water Heater

Figure 18 shows the power consumption of a water heater under normal conditions and with minor (7.5%) and major (35%) errors. The water heater’s power consumption pattern is influenced by its heating phase, where errors cause substantial increases in energy usage.

Table 8. Overview of the experimental dataset for a water heater.

Statistic	Normal Data	Anomaly Data (Minor)	Anomaly Data (Major)
Mean (W)	253.893	267.699	332.963
Median (W)	0	0	0
Standard Deviation (W)	1007.974	1033.175	1143.125
Variance (W2)	1,016,012.6176	1,067,451.803	1,306,737.0230
Skewness	3.725364	3.60718	3.1474
Kurtosis	11.909	11.0402	7.9272

presents the statistical features extracted from the 24-h usage of a water heater. With minor errors, the mean power consumption increases from 253.893 W (normal) to 267.699 W. The standard deviation and variance also increase, indicating a moderate rise in energy usage due to these errors. The skewness and kurtosis values decrease slightly, indicating a minor change in the distribution of power consumption patterns. For major errors, the mean power consumption increases significantly to 332.963 W. The standard deviation and variance also increase further. The skewness and kurtosis values further decrease, reflecting a more pronounced shift in the distribution.

For the water heater, Levene’s test and the Shapiro–Wilk test were conducted to ensure the reliability of the analysis. Levene’s test for the water heater showed no significant differences in variances between normal and minor error conditions ($F_{static}$ = 0.765, p = 0.3815) and significant differences in variances between normal and major error conditions ($F_{static}$ = 19.755, p = $9.131$ ∗ $e^{-6}$). These results indicate that the variability in power consumption is not consistent across these conditions. The Shapiro–Wilk test results for normal operation ($F_{static}$ = 0.991, p = 0.771), minor error ($F_{static}$ = 0.993, p = 0.609), and major error ($F_{static}$ = 0.979, p = 0.612) conditions confirmed that the data are normally distributed (p > 0.05).

The exported dataset for the presented home appliances, namely, fridge, washing machine, dishwasher, dryer, and water heater, as well as for the minor, major, and specific errors, is available in the GitHub repository https://github.com/AlexisPapaioannou/Power-Consumption-Anomaly-Dataset. The data is organized in csv format files, with comprehensive labels provided for the anomaly power consumption patterns. This dataset serves as a valuable resource for researchers and practitioners in the field, facilitating the development and testing of robust anomaly detection algorithms and predictive maintenance strategies [53].

5. Conclusions

This paper introduced a novel method for simulating error patterns in the power consumption of various home appliances to be used as ground truth for anomaly detection methods. Different home appliances were used, namely, fridges, washing machines, dishwashers, dryers, and water heaters. For each home appliance, an extensive analysis of the different phases that occur during an operational cycle was presented. Different algorithms were developed to create simulated anomaly data for each different type of error for each device. Additionally, an extensive analysis of detected anomaly patterns for each home appliance was presented.

The analysis of the simulated results indicates that the proposed method can accurately replicate the impact of different types of malfunctions on power consumption patterns. This capability is crucial for developing predictive maintenance strategies that can detect and address potential issues before they escalate, thereby improving the reliability and efficiency of home appliances. The statistical analysis of the simulated data highlights the significant shifts in power consumption patterns caused by various error types, underscoring the importance of effective anomaly detection systems in smart home environments.

Regarding future work, more home appliances and more specific errors will be added to cover an even broader spectrum of appliances and error types. Additionally, the challenging task of enhancing the granularity of the simulation to capture more detailed consumption patterns and anomalies will be attempted.