## 1 Introduction  
With the emergence of smart grids and the increase of electricity demand, home energy management system (HEMS) is attracting increasing attention as an important part of demand side of management. HEMS is able to improve the implementation of the distributed energy resource and shift the demand to off-peak periods [1]. At present, monetary incentives are the major driving force for residential consumers to participate in household load management. Flexible pricing schemes, such as critical peak pricing (CPP), time-of-use (TOU), and real-time pricing (RTP), are designed by utilities to encourage users to actively respond to demand side management [2; 3]. Responding to the various pricing schemes, load scheduling plays an essential role in HEMS [4; 5], which optimizes working schedules for household appliances to reduce the electricity bill and improves the power consumption efficiency while satisfying the consumers' comfort constraints [6]. Domestic electric water heaters (DEWH) are considered as well suited for demand side management, because of their high nominal power ratings combined with large thermal buffer capacities [7]. In addition, DEWH load accounts for a considerable share of the whole household load. In USA and Japan, DEWH load contributes as much as 17% and 27% to the total household electrical energy consumption [8; 9]. Therefore, the scheduling strategies for DEWH load are vital for HEMS to help consumers to automatically obtain optimal DEWH load schedules. With such background, several valuable works have been done on DEWH load scheduling. There are rich scheduling algorithms that are aimed to minimize the electricity payment under different rate structures, and various strategies for DEWH scheduling have been studied extensively. In [10], three consumer strategies were studied to schedule DEWHs under dynamic pricing, including timed power interruption, price-sensitive thermostat, and double period setback timer. Furthermore, a series of scenarios with different set points of water temperature in tank (between 120 and 140 \({}^{\circ}\)F)were analyzed to study the relevance of electricity cost to set points. In [11], DEWH load was deemed as an example of thermostatically controlled household loads, and scheduled through a linear-sequential-optimization-enhanced, multi-loop algorithm. A transactive control strategy was utilized to schedule the DEWH load iteratively according to the sorted day-ahead forecast price. Though the proposed optimization strategy is indeed simple, the created schedules are not necessarily cost-efficient. Given this deficiency, Reference [12] put forward a traversal-and-pruning (TP) algorithm to address the optimization problem of DEWH load scheduling. To handle the curse of dimensionality of the solution tree, a novel method named \"Inferior Pruning\" was proposed to traverse to the superior node's sub-nodes. Simulation results indicated that approximately 20% energy cost was reduced with the same comfort level as that of the reference method. However, in the above literatures, all of the relevant parameter values in household load scheduling are considered to be forecast completely accurately, which is not the real case in practice. Generally, forecast errors are inevitable for some parameters such as weather conditions, consumer behaviors, etc. [13]. With such a background, studying methods to tackle the uncertain parameters caused by forecast inaccuracy is of great significance for household load scheduling. In [14; 15], a two-stage stochastic optimization approach was introduced to tackle the uncertain household load scheduling. The decision variables were separated into two types of sub-variables named here-and-now variables, which are asked for decision-making at a given time and wait-and-see variables that can be determined later when enough information is available.