#### 1.1. General Context

Hybrid Renewable Based Systems (HRS) are promising alternatives for electricity supply in remote areas and are also known as stand-alone microgrid systems [

1]. The objective of the stand-alone microgrids is to provide energies based on green technologies to people in remote areas, permitting them to augment their productive capabilities and enhance their quality of life [

2]. This is possible to implement, thanks to the advances in renewable energy technologies that have allowed the installation of power generations in remote areas, which in turn benefit and cover non-interconnected areas.

Stand-alone microgrid systems can include different types of energy sources (photovoltaic and wind) [

3], storage systems (battery banks and supercapacitors) [

4], and loads. These elements can be connected through alternating current (AC) or direct current (DC) grids. In Reference [

5], a comprehensive review of AC and DC microgrids was presented. DC grids are preferred because they have a higher power density than AC grids; in addition, they do not require synchronization and incur only minor losses due to the skin effect [

6].

Figure 1 shows a representative scheme for an HRS where the elements are connected through a DC grid.

Note that the DC network concept comprehends an extensive range of applications, from high-voltage (examples are given in [

7,

8]) to low-voltage levels (examples are provided in [

9,

10]).

In addition, DC networks are especially attractive in control applications since the droop controls of reactive power and frequency disappear in these networks, making it easier for power flow control through lines in high-voltage levels or voltage regulation in low-voltage usages [

11]. Another important aspect of the DC network is the possibility of providing service to rural or remote areas with renewable source and energy storage devices, as depicted in

Figure 1; this helps improve the living conditions in those areas. In Reference [

10], a nonlinear controller for a typical configuration of a rural microgrid was presented and in [

12] a comparison between DC and AC microgrids implementations for rural social-economic development was performed.

#### 1.2. Motivation

The interconnection of each part of the system with the DC grid is achieved using power electronics converters, which are responsible for managing the power flow among the sources, storage systems, and loads. In Reference [

13,

14], examples of the use of the power electronics converters for such interconnections were described. The objective of the power flow control in a hybrid system is to satisfy the energy demand on the loads, maximize the energy extracted from renewable energy sources, and use storage systems efficiently.

The converter is entrusted with controlling the AC voltage applied to the loads, which is usually a DC-AC converter with an LC output filter [

15]. This converter can be single-phase or three-phase, depending on the load type. Meanwhile, its control regulates the amplitude and frequency of the output voltage based on a DC voltage applied on the input, which can then be controlled for the remaining HRS [

16,

17,

18].

The control strategy proposed in this research is motivated by the necessity to have robust and stable control methods for providing sinusoidal voltages in remote areas where conventional power systems are nonexistent [

19]. This entails that the opportunity to provide electrical service is by interfacing renewable energy resources (mainly wind turbines and photovoltaic plants) with power electronic converters that can regulate voltage and frequency by tracking sinusoidal references [

20]. The approach that uses sinusoidal references is different from a conventional emulation of synchronous generators via virtual inertia control [

21] since it is recommended for weak grids with large frequency variations. In these control schemes, active and reactive power measurements are used to define the frequency and voltage references [

20]. Nevertheless, the proposed controller in this paper is focusing on supplying electrical service to linear and non-linear loads directly interfaced with VSCs, which implies that the measures of active and reactive power are not efficient in regulating the output voltage. For this reason, our aim is to have a direct voltage control strategy based on trajectory tracking via passivity-based control approach with experimental validations, allowing supporting three-phase balanced voltages in passive and switched loads.

#### 1.3. Brief State-of-the-Art

In specialized literature, several strategies have been developed for the control of DC-AC converters. Due to their simplicity, the most widely used approaches are based on classic linear controllers, which are proportional-integral (PI) controllers [

22]. Even though these strategies are the most used, they cannot guarantee the stability of the system. Additionally, they do not perform well away from the point of operation as in the case of non-linear loads. [

16]. Therefore, advanced strategies have been developed to address the poor performance of classic controllers. In Reference [

23], a feedback linearization control method was proposed based on a power-balance model between the converter and the load. This method improves the performance of linear and nonlinear loads, but the selection of gains is critical. In Reference [

24], a current control algorithm for uninterruptible power supplies based on PID compensator was presented. [

25] showed the reduction of voltage distortion caused as a result of slowly varying harmonic currents that use synchronous-frame harmonic regulators.Reference [

26] describes an integral resonant controller of the output voltage management arrangement in a three-phase VSI. Reference [

27] presented a model predictive control for output voltage regulation of a three-phase inverter with output LC filter feeding linear and nonlinear loads. In addition, authors in [

28] use the same control strategy for a single-phase voltage source with linear and nonlinear loads. Despite previous works demonstrating good performance of their objective controls, none of them can guarantee the stability of the system.

On the other hand, the application of passivity-based control (PBC) techniques to power converters has the advantage of providing stable closed-loop controllers with good dynamic behavior. In Reference [

16], an interconnection and damping assignment (IDA-PBC) approach was proposed to regulate the output voltage from a DC-AC converter and a comparison was also made with classic controllers. The results of [

16] demonstrated the good performance of the proposed controller, even when a nonlinear load was considered. An adaptive robust control method for a DC-AC converter with high dynamic performance under nonlinear and unbalanced loads was also proposed by [

29]. In both of these methods, the stability is ensured by the passive properties of the controlled system [

30]. The problem with the PBC controllers applied to power converters is that the control laws depend on the system parameters and, so, stable-state errors occur when these parameters vary. The errors caused by variations in the system parameters can be eliminated with different techniques; for e.g., a dynamic extension with an integral action was proposed by [

16] but at the expense of increasing the complexity of the system.

PI-PBC controllers have been proposed to combine the advantages (simplicity and robustness) of PI-based designs with the typical stability analysis based on the Lyapunov theorem employed in passive strategies. These controllers have been used in power converters for several applications [

31,

32,

33,

34].

Authors in [

31] have presented the general basis of the PI-PBC theory applied to power electronic converters (switched systems). These authors demonstrate that with PI gains in a Hamiltonian representation of the averaged dynamic of the converter is possible to provide constant direct current–voltage to linear loads. Simulation and experimental results demonstrated that when VSCs are used in conversion mode (sinusoidal input to DC constant output), the PI-PBC method guarantees asymptotic stability if the load is completely linear (i.e., resistive). Observe that the VSC was operated with sinusoidal voltage imposed on the AC side to generate constant DC voltage. This is a different case, compared to the approach presented in this paper as we work with constant DC voltage provided by a combination of batteries and renewables to support three-phase balanced voltage in linear and nonlinear loads, guaranteeing stability conditions in the sense of Lyapunov. In reference [

32], a general design using the PI-PBC method was presented for tracking trajectories in power electronic converters (sinusoidal or constant references) if they are bounded and differentiable (i.e., admissible trajectories). The stability in closed-loop is ensured via Barbalat’s lemma. The authors of this paper validate their control design in an interleaved boost and the modular multilevel converter, including simulation and experimental validations. Note that the first converter works with AC input to provide a constant DC output, while the second one generates single-phase voltages in linear loads considering a constant DC input. This implies that the application of the developed PI-PBC method is different from our approach since we work with the isolated network applications to generate three-phase voltage signals in linear and nonlinear loads. The authors in [

33] presented a methodology based on dynamic power compensation of active and reactive power in transmission systems considering superconducting coils integrated via a cascade connection between DC-DC chopper converter and the VSC. The control for this system is developed with PI-PBC, guaranteeing stability in closed-loop. The main aim of this paper is to compensate subsynchronous oscillations in power systems when faults occur in the power grid. Note that the proposal of these authors works with the VSC connected to the grid by controlling active and reactive power flow; while in our approach, the power grid is non-existent and the objective is to provide voltage service to isolated loads, i.e., we generate the power system node with constant voltage and frequency via PI-PBC design. In [

34], standard passivity-based control design for integrating renewable energy resources in power systems was developed. The main idea of [

34] is to provide a stable control design via PBC, which is made via energy functions using a Lagrangian formulation. Additionally, it is assumed that the wind generator would be connected to the power grid. This implies that the electrical network supports the voltage on the AC side of the converter. For this reason, the authors of this study focused on active and reactive power control and not on the three-voltage generation for isolated power applications as the case studied in our contribution.

In Reference [

35], a general control design of controllers for single-phase network applications was presented via interconnection and damping assignment PBC and PI-PBC approaches. In this work, the authors considered isolated power grids composed of batteries, wind turbines, photovoltaic plants, and energy storage devices composed of superconducting coils and supercapacitors. The main contribution in [

35] was to demonstrate stability in single-phase networks under well-defined load conditions. Even if this research uses isolated systems by applying PI-PBC control, it is different from our contribution since, in our work, the grid has a three-phase structure and the loads are strong, nonlinear loads (switched devices), which were not considered in [

35]. Note that in [

36] the initial design based on PI-PBC and IDA-PBC was complemented with modifications on the controller structure to integrate renewables in single-phase networks. In addition, the difference with our approach is that the authors do not present any experimental test that validates their simulation analysis.

Authors in Reference [

37] presented a general stability analysis for single-phase networks feed-through power electronic converters considering constant power load. This analysis was performed assuming a Hamiltonian representation of the system and the perfect operation of the controllers that manage the power flow between the distributed energy resources and the grid. The authors of this work do not mention how this approach is extensible to AC grids with strong nonlinear loads as the case study in our proposal.

Even if controllers based on PI-PBC have been proposed for controlling power, electronic converters in single-phase and three-phase applications. In this paper, we focus on the problem of the voltage generation in three-phase nonlinear loads located in isolated areas by deriving the PI-PBC approach from the classical IDA-PBC method [

16], which has not been reported in the scientific literature yet. In addition, our work contains multiple simulation scenarios and some experimental validations that validate the proposed approach, demonstrating its easy implementation in real-life operative cases that combine renewables, batteries, power converters, and nonlinear loads.

It is important to mention that it is necessary to employ optimal tuning of the PI gains so that PI controllers (including classical PI and PI-PBC approaches) perform excellently [

38]. Active/passive tuning methods have been reported in the scientific literature. In Reference [

39], it was presented an interactive tool for adjusting PI controls in first-order systems from a graphical point of view for first-order systems with time delays, numerical results confirm the efficiency of the tool developed in comparison with other literature reports. In Reference [

40], an algorithm for the PID controller based on the gain margin and phase margin concept was presented. However, the controller parameters depend on a single parameter, these parameters are subjected to the desired phase margin, and a minimum required gain margin constraint. The main advantage of these tuning an approach with respect to previous works is that it is easy to implement applicable to any linear as well non-linear model structures. Authors of [

41] have presented a simple method to design PI controllers in the frequency domain by proposing an optimization model with constraints. This method uses a single tuning parameter, defined as the quotient between the final crossover frequency and the zero of the controller. This adjusting procedure maximizes the controller gain by considering the equality constraint on the phase margin and an inequality restriction in the gain margin. Numerical results confirm the effectiveness of this proposal in comparison with literature reports. Additional methods for tuning PI controllers have been reported in specialized literature, some of them are particle swarm optimization [

42], ant-lion optimizer [

43], genetic algorithms [

44], and so on. The main feature of these metaheuristic optimization methods is that they work with the minimization of integral indices to find the optimal set of control gains by using sequential programming methods [

45].

**Remark** **1.** The selection of control gains is an important task in the design of PI controllers in power converter applications. These methods can be passive or active approaches that work with optimization models or desired performances [46]. Nevertheless, in this research, our focus is on presenting a simple controller based on the properties of the passivity theory combined with classical and well-known PI actions to generate ideal three-phase voltages for non-linear loads in isolated areas. This implies that the focus in the grid performance with load variations and no optimal adjusting of the control gains. In this sense, we employ a basic tuning method based on the root locus design approach [47]. #### 1.4. Contribution and Scope

In the present study, a PI-PBC controller is proposed for regulating the amplitude and frequency of the output voltage in a three-phase DC-AC converter with an LC filter, providing a well-defined sinusoidal service to linear and nonlinear loads by transforming the DC signal from the transmission/distribution network to local loads [

48].

The main contribution of this research in the literature reports about the control of VSCs for feeding isolated three-phase loads can be summarized as follows:

- ✓
A passivity-based control design that is easily implementable with the main advantages of the classical PI controllers that allows tracking a sinusoidal trajectory by transforming this into a regulation problem. The proposed PI-PBC design also allows guaranteeing stability conditions based on the Lyapunov theory by applying the properties of the Hamiltonian energy models.

- ✓
The proposed controller can maintain objective controls, which are to regulate constant voltage amplitude and constant frequency although the test system feeds a non-linear load, demonstrating the generation of a robust three-phase balanced signal. This is achieved by avoiding the use of classical phase-locked loops embedded in virtual synchronous emulations that emulates inertia properties in converters.

- ✓
The experimental validation in a laboratory prototype with a realistic model of the system include switching effects, losses, and a detailed transistor model to feed passive loads and nonlinear ones.

In addition, the performance of the controller under parametric variation is shown, and a comparison with the classic PI controller demonstrates the superiority of the proposal to mitigate the harmonic content produced by non-linear loads.

Regarding the scope of this research, it is important to mention that in the control design as well as in the simulation, experimental validations will be considered unique voltage source converters that are forced to work as an ideal voltage source to provide sinusoidal voltages to linear and nonlinear loads. For doing so, we consider that the DC side of the converter is fed by a strong DC network (transmission/distribution DC grid) or by a combination of renewable energy resources and batteries [

19,

28]. In addition, to determine the amount of instantaneous power absorbed by the load (linear and nonlinear), it is considered that there is a current measure at the load side which is important since the amount of current provided by the converter is a linear function of the load consumption. This implies that the existence of this measure is indispensable when no load estimators are implemented, as in the case studied in this research. Note that the implementation of load estimators could be considered for future work since only a few studies have been reported in the scientific literature with experimental validations.