## 1 Introduction
### Related Works  
The evolution and optimization of HEMS have seen substantial progress over the past few decades. The foundation for the modern smart grid was laid by the electrical power research institute (EPRI) in 1998, which researched a sophisticated interactive system designed to create a highly reliable and fully automated grid in the United States, laying the foundation for the U.S. smart grid [24]. Several studies have explored various approaches to optimize HEMS. In [25], a study integrated a smart thermostat with a HEMS using MILP for day-ahead load scheduling to reduce costs and enhance DR and PV self-consumption. The fuzzy logic-based thermostat managed air-conditioning efficiently. A simulation showed a 53.2% cost reduction under the TOU rates of Turkey and a 24% AC cost reduction compared with conventional thermostats. An energy management system using reinforcement learning (RL) and fuzzy reasoning (FR) was proposed [26]. It optimally scheduled appliances, considering user preferences, and showed significant cost reductions and consumption smoothing. The system employed Q-learning for optimal decision-making, shifting appliance usage to off-peak hours. A MATLAB/Simulink interface demonstrated the DR scheme with smart appliances and energy sources. A HEMS using deep Q-learning (DQN) and double deep Q-learning (DDQN) for scheduling home energy appliances was proposed [27]. These model-free algorithms help customers reduce electricity consumption in a dynamic environment. Tests show that DDQN is more effective than DQN for cost minimization. Detailed analysis and comparisons with particle swarm optimization (PSO) validated the performance. Effectiveness was confirmed using a real-world database combined with a household energy storage model. In [28], their study explored DR in household appliance usage by analyzing factors affecting electricity usage behavior. It utilized K-means clustering to categorize households and estimate daily electricity consumption patterns, crucial for effective DR strategies in Greater Accra, Ghana. The study ensured clustering quality using the Silhouette Score and principal component analysis, integrating qualitative and quantitative data. Consumer behavior surveys further refined insights into appliance usage trends. Research has also focused on addressing the mismatch between energy supply and demand through flexible load management, achieving notable reductions in electricity costs and improvements in renewable energy utilization [29]. The study utilized a genetic algorithm to minimize electricity costs and improve renewable energy use, showing a 22.5% cost reduction and a 16.5% increase in renewable energy utilization. In [30], the integration of attention networks into HEMS was explored to enhance energy consumption optimization. By using the AMpds2 dataset and evaluating time series forecasting models, the study found that transformers improved forecasting accuracy by 4%. The implementation used Python 3.2 and the matplotlib library for visualizing results. In [31], the day-ahead operation of a smart residential distribution electrical grid was demonstrated using a two-stage multiple-criteria decision-making model. The study optimized energy consumption costs, power losses, and demand-side comfort by employing load shifting and interruption strategies. Improved sunflower optimization (IFSO) and TOPSIS were utilized, demonstrating effectiveness on a 33-bus distribution network. Studies on shared energy storage and enhanced DR have addressed issues related to the low utilization of energy storage caused by uncertainties in energy sources and load demands, demonstrating improved energy storage utilization and operational economy [32]. By expressing source-load uncertainty with interval numbers and analyzing DR mechanisms, the model optimized source-grid-load-storage interactions. Sensitivity analysis on the linearized objective function under dual-side uncertainty improved result accuracy.