# Robust Optimization for Household Load Scheduling with Uncertain Parameters
## 1 Introduction
Due to the complexity of impacting sources in the forecast model and randomness of human behaviors, some forecast parameters, especially like the ambient parameters and customer behavioral parameters, are still not accurate enough to represent the real environment. Meanwhile, most forecast methods need a large amount of historical data which is difficult or expensive for household customers to obtain. Especially, the stochastic programming accounts for a large part of the uncertain optimization methods. Since the characteristics of heuristic algorithms are widely used in the stochastic programming, it is inevitable that the results will be trapped in local optima and make the iteration rather time-consuming. Differing form the above studies, this paper applies a robust optimization method to tackle the uncertainties in household load scheduling. In 1973, [PERSON] firstly gained the solutions under the worst situation of uncertain parameters through linear robust optimization [15]. To decrease the conservatism of linear robust optimization, [PERSON] et al. [16] proposed the adjustable robust optimization counterpart. [PERSON] and [PERSON] [17; 18] extended the robust optimization by introducing an adjustable factor to reflect the different choices from the uncertain parameters set. The robust optimization proposed by [PERSON] has some advantages in handling the uncertainties in parameters: through modeling the uncertain parameters varying in a given uncertainty set, it avoids complex random variables which are subjected to the probability distribution or fuzzy membership function, and it maintains the linearity of the problem so that the global optimal solution can be obtained by a linear programming solver. Additionally, based on setting different values of the factor, it can flexibly control the conservative levels of the results. Based on these, quite a few works [19; 20; 21] pay significant attention to robust optimization in HEMS. Thus, this robust method is applied in HEMS to solve the scheduling problem with uncertainties. In this paper, a home energy management system equipped with photovoltaic generation and an energy storage device is established. A household load scheduling with uncertain parameters is formulated to minimize the electricity bill of customers and operates under the comfort constraints. Specifically, the main contributions of this paper can be summarized as follows:  
1. Two typical examples of uncertain parameters, outdoor temperature and hot water demand, are modelled as uncertainty sets, based on which household load scheduling problem with uncertainties is formulated. For researching the uncertainties, the uncertain parameters are presented in the form of interval numbers. A robust optimization method is applied to deal with the uncertainties in the comfort constraints. The robust counterpart transformation is the key component of robust optimization. Via deducing the robust counterpart, the original scheduling problem is transformed into a mixed integer linear programming problem, of which the global optimum can be found by mature tools, such as CPLEX. The proposed method avoids the time-consuming iterations in many other uncertain optimization methods. 3. An uncertainty analysis that quantifies the violation degree of the comfort constraints is designed and conducted to test the proposed method. The results show that the complete robust schedules is able to guarantee that the comfort constraints will not be violated. Moreover, schedules with different robust levels can be obtained to make a trade-off between the comfort violation and electricity payment. This paper is organized as follows: Section 2 introduces the mathematical model of the home energy management system, including the uncertain parameters, objective function, and constraints; Section 3 introduces the robust optimization method and completes the derivation of the robust counterpart; and, finally, the simulation results are presented and analyzed in Sections 4 and 5.