## 1 Introduction
In the context of optimal thermal comfort, during the last two decades a lot of effort has been devoted to investigating Model Predictive Control (MPC) strategies [12]. The main advantage provided by MPC is that it is possible to take into account the set-point scheduling and the weather forecasts and it is easier to deal with linear and nonlinear constraints. Thus, because of its flexibility, MPC has been widely used in the building sector to manage the trade-off between, typically, energy usage and comfort level [13; 14; 15; 16]. In other works, the use of MPC is proposed to minimize the energy consumption while maintaining the indoor thermal comfort at an adequate value by constraining the temperature value between two bounds [17]. In any case, it must be considered that a centralized MPC strategy could have problems when the size of the system increases, as the time interval and the computational effort required to solve the optimization problem might become too large. To address this issue, the application of a distributed or decentralized MPC has been proposed, but the main drawback in this case is that the overall optimization problem is decomposed in many smaller problems and it is likely that a local minimum is found. For example, a decentralized predictive control strategy to perform a temperature set-point following in a nine rooms building has been applied in [18]. The methodology is quite complex because, firstly, a partition scheme of the building needs to be done and therefore many different lumped parameters models need to be evaluated. Secondly, an order reduction of the model is applied to all the different partitions. An improvement to this kind of decentralized control strategy comes from the distributed predictive control approach where the partitioned controllers exchange information between each other to obtain the minimum of their own objective function by also knowing the status of the other subsystems. The different performance obtained by centralized, decentralized and distributed predictive control applied on a small building has been investigated in [19; 20]. It has been concluded that the distributed control strategy reduces the computational burden with respect to the centralized one by obtaining in any case a satisfactory performance. In general, the MPC strategy has many advantages and for this reason it has been widely used in many different applications. However, a predictive controller needs to solve the optimization problem at each sampling period and, in the case of a thermal system such as a building, due to its big time constant, it should consider a quite long prediction horizon to be effective. Furthermore, if the size of the problem increases it can be difficult to find the optimal solution in a single sampling period. In this paper, which is an extended version of [21], we consider the previously described situation where different set-point values are imposed by the user for the different rooms and for different time intervals of the day and we propose a different approach, based on genetic algorithms [22], to the problem of determining the optimal reference signals for the rooms of a residential building to satisfy the user requirements (in terms of a given temperature for each period of a day) by minimizing the energy consumption. The genetic algorithms (GA) approach has been widely used in the building sector; however, it has been mainly applied for the optimization of building design. In this case, a multi-objective approach is used to design the building thermal parameters to obtain a significant energy consumption reduction, or to place and choose the correct HVAC equipment [23; 24]. A more interesting case is when derivative-free algorithms, such as GA or particle swarm algorithms, are used to minimize the energy consumption of HVAC systems by optimizing their operations. For instance, in [25] a multi-objective approach is employed to control an air-handling unit, so that a potential energy saving up to 30% is obtained.