## 1 Introduction  
With the development of the economy, the share of electric power consumption of residential customers has dramatically increased, especially in developed regions [1]. In order to improve customers' energy efficiency, home energy management systems (HEMS) have been widely studied [2; 3; 4]. Generally, the HEMS can schedule electric appliances' operation status (ON/OFF status or comfortable inside temperature) according to the setpoints of customers [5]. However, these methods depend on the customers' setpoints, and cannot automatically and intelligently help people make optimal decisions. In recent studies, some researchers have proposed optimal energy strategies, which schedule the electric appliances to minimize energy cost while maximizing customer comfort satisfaction [6; 7]. In order to realize this objective, two key problems should be tackled. The first is how to quantitatively evaluate customers' satisfaction, considering both energy-saving performance and the risk of comfort loss. On the other hand, we need to study how to optimally schedule the electric appliances' operation status based on the satisfaction evaluation indexes [8; 9]. As for the first problem, many recent researchers have evaluated customer comfort from the perspective of indoor environmental quality (IEQ) [10]. For instance, [11; 12] solved the energy-saving problem in buildings through structural design and feedback techniques, respectively. Although theses IEQ-based methods are able to improve comfort and energy saving, they focus more on inhabitants' behaviors and habits, and the planning and optimization before the fact. In a real-time energy-saving optimal control field, utility functions have more usually been used in recent research to evaluate customers' income when electricity is consumed. [13; 14] used a utility function that has a positive marginal return and a diminishing marginal rate of return from an economic viewpoint. In general, a logarithmic function, a diamond growth model, and a quadratic function are all suitable choices [15]. However, customers are assumed to be entirely rational in conventional utility functions. Utility functions are a concept from economic theories, while electricity is not wholly the same as general goods. Besides utility functions, some physical indexes are also adopted to evaluate the satisfaction of customers in electricity consumption. For instance, [16] used the real temperature to assess the performance of a chiller, while the state of charge was used to control the energy storage systems [17]. Physical indexes can only be used for one or two individual devices, and are not universal in residential energy-saving because there are various types of appliances in HEMS. To tackle this problem, conventional utility functions are improved by considering customers' bounded rationality, based on concepts from behavioral economics [18]. Accordingly, in our previous work, the utility function of electric appliances was combined with prospect theory for the sake of quantifying the customers' bounded rationality [19; 20]. However, only several specific electric appliances were studied. In this paper, electric appliances are not categorized from a new perspective. We utilize the standard categories of time-sensitive and temperature-sensitive appliances, which cover nearly all the efficient electric appliances of residential customers. Based on our new clustering method, we extend the utility function into a new concept--energy-saving cost--while considering prospect theory to better depict the real behavior of customers. In conclusion, the proposed energy-saving cost is able to measure the satisfaction of a customer before and after optimization. Based on this measurement, we are able to formulate the optimization problem for energy saving, where the energy-saving is used as a constraint to make sure a customer's comfort will not be reduced. Besides the satisfaction evaluation problem, how to optimize the appliances' commitments is the second issue in energy-saving programs.