## 1 Introduction
Regarding the first part, the optimal EMS designed is an MPC-based EMS that implements the formulation of a mixed integer linear programming (MILP) problem. It has been programmed in Matlab/Simulink. For the formulation of the MILP, the techno-economic model developed by the authors in [PERSON] et al. [15] has been used. This model represents an advance over the models available in the literature. The reason for this is that it is more precise and realistic, because it includes physical phenomena that are known to be essential for the correct evaluation of the economic performance and that other models have not taken into account: fuel cell degradation, auxiliary energy losses, heat losses in the thermal energy storage (TES), etc. Regarding the second part, there is no previous work in the literature that has examined what the appropriate prediction horizon is for the optimal operation of micro-CHP systems. Regarding the third part, there are several studies that have evaluated the economic performance of an optimal EMS for micro-CHP, with different purposes [17; 18; 19; 20; 21; 22; 23; 24; 25]. One of these purposes is to determine how much economic benefit the optimal EMS achieves compared to rule-based EMSs. In effect, it is precisely against rule-based EMSs that the optimal EMS must be evaluated and compared, since these are the EMSs currently in use and the ones the optimal EMS would replace if it surpassed them in economic performance. In this respect, the main deficiency of these works is that none of them include in their comparison the four rule-based EMSs that the authors showed to be relevant, namely (1) heat-led, (2) electricity-led, (3) heat-and-electricity-led, and (4) electricity-led switch off in summer (see [PERSON] et al. [15]). Because of this, there is still no well-established result on how much profit an optimal EMS achieves compared to rule-based EMSs. Finally, regarding the fourth part, only one study, [PERSON] et al. [17], has investigated the behavior of an optimal EMS for micro-CHP systems and identified optimal operating patterns. This fourth task, however, remains of interest and needs to be addressed again, despite having been carried out by a previous work, for three reasons. First, in [PERSON] et al. [17], the micro-CHP system was based on an SOFC (solid oxide fuel cell), not a PEMFC. There are enough differences between these two technologies to suggest that the optimal EMS behavior might be different in each case. The study by [PERSON] and [PERSON] [25] provides additional evidence in support of this hypothesis. Second, the study by [PERSON] et al. [17] was limited to the United Kingdom, so its energy prices space is considerably smaller than the one in Europe. A larger energy prices space could reveal new optimal behaviors not detected by [PERSON]. The results of [PERSON] et al. [15] represent strong evidence in favor of this hypothesis. In effect, they showed that a change in the energy prices causes the best rule-based EMS to change. Third, [PERSON] et al. [17] say nothing about how stack degradation influences the behavior of the optimal EMS. Stack degradation is a fundamental phenomenon that greatly affects the performance of the micro-CHP system throughout its lifetime; see [PERSON] et al. [15] (Section 2.1). Furthermore, [PERSON] et al. [15] showed that changes in the stack degradation level also cause the best rule-based EMS to change. This fact suggests that the behavior of the optimal EMS is very likely to change with the stack degradation level. In summary, regarding the optimal EMS behavior (fourth part or task), the knowledge gap is large. In effect, the following are not known:  
* Whether or not the results of [PERSON] et al. [17] will be the same for a PEMFC-based system. * If their conclusions remain valid for a wider energy price range, which is what exists in Europe. * What effect stack degradation has on the optimal EMS behavior. Table 1 compares this study with five previous studies.