## 2 Electric Appliance Utility Function and Energy-Saving Cost
### Energy-Saving Costs  
Under the condition of guaranteeing user satisfaction as much as possible, energy-saving optimization for electrical appliances refers to adjusting the operation status to the most energy-saving status. In order to quantitively evaluate the comfort loss risk for the customers in the energy-saving program, this paper proposes a new concept: The energy-saving cost. The energy-saving cost is defined as the difference of the utility function value before and after energy-saving optimization, shown as Equation (4). \[\hat{C}_{i}(t)=U_{i}(t)-\hat{U}_{i}(t). \tag{4}\]  
In this equation, \(\hat{C}_{i}(t)\) denotes the energy-saving costs of appliance \(i\), accepting a certain optimization strategy at the time slot \(t\). \(U_{i}(t)\) is the appliance utility value before executing the procedure, and \(\hat{U}_{i}(t)\) is appliance utility value after implementing the strategy. If energy-saving costs are positive, it indicates that the energy-saving strategy will weaken user satisfaction,but if energy-saving costs are negative, it represents that user satisfaction will not be damaged by optimization. Since the calculation of time utility is based on probability, misoperation cannot be avoided entirely. Once misoperation occurs, user satisfaction will sharply drop to near zero. Thus, for time-type appliances, a further definition of energy-saving costs is proposed to avoid misoperation for those operations that may lower utility value, shown as Equation (5). \[\hat{C}_{i}(t)=\left\{ \begin{array}{ll}U_{i}(t)-0&\gamma U_{i}(t)\geq\hat{U} _{i}(t)\\ U_{i}(t)-\hat{U}_{i}(t)&\gamma U_{i}(t)<\hat{U}_{i}(t)\end{array} \right.. \tag{5}\]  
In this equation, \(\gamma\) is the energy-saving risk control factor, which can affect the misoperation risk of the energy-saving strategy. If \(\gamma\) equals 1, users tend to refuse those operations which may lower appliance utility. A smaller \(\gamma\) means that users are more willing to take the risk of misoperation. On the contrary, the larger \(\gamma\) is, the less willing they are to accept misoperation. Then, considering the bounded customer rationality, prospect theory in behavioral economics [18] is introduced into the energy-saving costs to evaluate the relationship between energy-saving and user satisfaction. The energy-saving costs evaluation method is shown as Equation (6), which is a mapping of the energy-saving costs to the users' subjective feeling for energy-saving costs. \[C_{i}(t)=\left\{ \begin{array}{ll}\eta\cdot\hat{C}_{i}(t)^{\theta}&\hat{C}_{ i}(t)\geq 0\\ -(-\hat{C}_{i}(t))^{v}&\hat{C}_{i}(t)<0\end{array} \right.. \tag{6}\]  
In this equation, \(\eta\) is the sensitive coefficient for energy-saving costs, \(\theta\) is the risk aversion coefficient for utility damage, which belongs to \((0,1]\), and \(v\) is the risk preference coefficient for utility gain, which has the same range. The larger \(\eta\) is, the less willing users are to accept energy-saving strategies which will influence user satisfaction. Furthermore, it is worth noting that energy-saving costs include both instantaneous energy-saving costs and accumulative energy-saving costs. In the time dimension, changes of appliance operation status will influence the appliance utility instantaneously, so that instant energy-saving costs can accurately characterize the gain or damage of user satisfaction. This is defined as Equation (6). On the other hand, in the temperature dimension, changes in appliance operation status cannot affect the appliance utility instantaneously when the control instruction is issued. For instance, when changing the air conditioner operation status (such as by turning on, turning off, or adjusting the set temperature), the environmental temperature does not change instantaneously, but changes gradually. Not only the trend, but also the magnitude of the change, depends on the specific setting of the new mode. In this situation, accumulative energy-saving costs for a continuous period characterizes the impact of the energy-saving strategy on user satisfaction.