## 1 Introduction
This is partly due to the simplicity of designing linear models which translates into less computational efforts [30]. Many researchers have addressed the issue of controlling the air conditioning home systems through different control methods along the years [29; 31]. A more generalized approach for mixed-integer predictive control of HVAC systems using (mixed integer linear programming) MILP is presented in [32]. To improve the wind power utilization level in the distribution network and minimize the total system operation costs, [PERSON] et al. in [33] propose a MILP based approach to schedule the interruptible air-conditioning loads. In [18], [PERSON] et al. implement and validate a model predictive control (MPC) based supervisory controller which is designed to shift the heating and cooling load of a house in Toronto to off-peak hours. A hierarchical decomposition for economic MPC in large-scale commercial HVAC systems using a two-layer approach is proposed in [34]. A case study in which artificial neural network (ANN) models of a residential house located in Ontario, Canada are developed and calibrated with the data measured from the location is addressed in [19]. In [35], [PERSON] et al. present an experimental case study of a scenario-based MPC for HVAC systems. In [36] an experimental implementation of whole building MPC with zone based thermal comfort adjustments is presented. A prototype embedded system is developed in [37] to emulate an adaptive environmental control strategy algorithm to numerically determine an indoor comfort temperature for a real-time control in an air-conditioning system. A demand side response modeling with controller design using aggregate air conditioning loads and particle swarm optimization is presented in [38]. In [39], [PERSON] et al. present a DR strategy to address residential air-conditioning peak load in Australia. An automatic AC control with real-time occupancy recognition and simulation-guided model predictive control by using Raspberry Pi 3 is presented in [40]. In [41], [PERSON] et al. present a mono-objective genetic algorithm (GA) that minimizes global costs for space conditioning for cost-optimal building design integrated with the multi-objective model predictive control. In [42], the problem of scheduling deferrable appliances and energy resources of a smart home is addressed considering a variety of sources applying a multi-time scale stochastic MPC. In addition, in [43], distributed energy resources scheduling problem of the set of smart homes (SHs) has been investigated considering their cooperation with their neighbors by applying a stochastic MPC. In [44], the implementation of demand response (DR) programs is investigated considering the nonlinear behavioral models of the residential customers. It is proposed in this paper to compare three control options for an AC unit, ON/OFF, PID and MPC. However, instead of adopting a linear power switch, a two level control signal interface for the MPC that modulates the bounded continuous set of manipulated variables to a discrete set of integers is used in this paper, as described in [45]. This AC unit controls the temperature of a room governed by six distinct DR electricity ToU rates. One ToU rate is an hourly price signal during 24 h while the remaining five ToU rates are the presently existing ones enforced by the Portuguese electricity retailer [46]. In the case of this model, the home is equipped with a PV solar panel. The selected setting for this study was the Portuguese city of Evora because it is a pilot city in a DR project--InovGrid [47]. In the model for this paper, an entire week of summer--July 2016--was investigated. The reason for picking this week is that the records confirm a noticeably high ambient temperature and a high solar irradiation. The overall purpose is to compare the overall weekly expense of each studied tariff option for every control method and in the end to reach an optimal solution. The rest of the paper is structured as follows.