A Direct Current (DC) microgrid is a concept derived from a smart grid integrating DC renewable sources. The DC microgrids have three particularities: (1) integration of different power sources and local loads through a DC link; (2) on-site power source generation; and (3) alternating loads (on-off state). This kind of arrangement achieves high efficiency, reliability and versatility characteristics. The key device in the development of the DC microgrid is the power electronic converter (PEC),

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... ince it allows an efficient energy conversion between power sources and loads. However, alternating loads with strictly-controlled PECs can provide negative impedance behavior to the microgrid, acting as constant power loads (CPLs), such that the overall closed-loop system becomes unstable. Traditional CPL compensation techniques rely on a damping increment by the adaptation of the source or load voltage level, adding external circuitry or by using some advanced control technique. However, none of them provide a simple and general solution for the CPL problem when abrupt changes in parameters and/or in alternating loads/sources occur. This paper proposes a mathematical modeling and a robust control for the basic PECs dealing with CPLs in continuous conduction mode. In particular, the case of the low voltage residential DC microgrid with CPLs is taken as a benchmark. The proposed controller can be easily tuned for the desired response even by the non-expert. Basic converters with voltage mode control are taken as a basis to show the feasibility of this analysis, and experimental tests on a 100-W testbed include abrupt parameter changes such as input voltage. Energies 2018, 11, 1130 2 of 20 wind generators with a rectifier and energy storage systems (batteries) with an energy management system, among others. A particularity of the DC microgrid with respect to the smart grids is the integration of different power sources joined through a DC link and connected to local loads. In addition, the energy source can be generated on-site in such a way that the microgrid achieves high efficiency, reliability and versatility [3] . Today, there are numerous electronically-managed power loads that can be connected to the DC-link of a smart grid, such as televisions, electric ovens, coffee makers, LED lamps, computing equipment, among many others. However, it is necessary to mention that power electronics is the key device in the development of a DC microgrid, mainly due to the use of PECs, which allow an efficient power energy conversion between sources and loads. There are different kinds of PECs that can be connected to a DC microgrid in order to reduce, invert or increase the desired/nominal DC-link voltage, i.e., buck, boost, interleaved boost, positive buck/boost, forward, fly-back, half bridge and full bridge converter, to name a few [4] . However, for this paper, a low voltage DC microgrid for building/residential applications similar to the one proposed in [5] is considered as a benchmark. This arrangement is divided into three sections as in the following description. The first one includes high power elements for common building facilities (greater or equal to 538 V and 10 kW), including photovoltaic panels, wind generators, fuel cells, among others. The second one relies on the medium power elements, which are usually found in kitchen appliances and laundry rooms ([230, 400] V, [0.4, 10] kW). It is necessary to mention that the current loading can remain similar to the 230-V AC system. However, the voltage can be increased up to 400 V if a size reduction is needed. Finally are the low power loads, which include devices located in bedrooms, living rooms and outdoor areas ([24, 48] V, less than 0.4 kW). It is well known that in a closed-loop scenario, the load can act as a constant power load (CPL) for its source, producing a negative-rate varying resistance (known as negative impedance), which destabilizes the system. Since the theoretical analysis complexity increases with this characteristic, the development of different approaches to ensure the stability of the closed-loop system has been encouraged. Traditional CPL compensation techniques rely on a damping increment by adapting the voltage level of the source or the load, adding external circuits or by using advanced control techniques. An interesting review of these approaches was reported in [6] , which classifies the CPLs into two groups, control approaches and auxiliary circuits/power buffer. In the first group, control techniques such as feedback linearization, sliding mode, pulse adjustment and pole placement were reviewed such that their main benefit is that they do not diminish the load performance. In the second group, the instability problem seems solved; however, an auxiliary circuit is needed, increasing the implementation cost and complexity; the overall system stability is omitted. Another attractive survey can be found in [7] in which a review of the state of the art in the performance and usual properties of DC microgrids connected to CPLs, stability approaches and compensation methods were described. Recently, a novel discrete-time modeling and an active stabilizer for PWM converters was proposed in [8] . This proposal allows performing the stability analysis for the DC distribution systems with nonlinear characteristics. The authors reported that the proposed discrete-time method could identify slow-scale and fast-scale instabilities and also a measure of the sensitivity to some parameter change. The robustness of the proposal to filter the parameters' variation was also studied. Finally, this proposal was numerically-and experimentally-verified. Nevertheless, a formal analysis of the CPL case is not reported. On the other hand, an active damping method was stated in [9] . In this paper, a variation of the CPL in the system by attaching a supercapacitor-based energy storage system was proposed. As a result, the CPL's instability effects can be decreased by virtually increasing the resistive loads. Numerical simulations and practical results were reported. The limitation of this proposal relies on the extra components needed such as a supercapacitor and a bidirectional converter; besides, the production of an energy consumption increase and a slow dynamics response are present.