The bibliography on control strategies for microgrids is extensive and addresses strategies for connected mode, islanded mode, and transition between modes [24], [25]. However, there is a clear and widespread tendency to organize control strategies in hierarchical structures [26], [27]. The primary layer is usually based on the well-known droop method [28]. This strategy produces a virtual inertia through the emulation of the synchronous generators' operation, allowing distributed generators to achieve a common frequency, which produces the proportional generation of the steady-state powers. However, this method requires the introduction of a deviation in the frequency of operation, which should be restored in the secondary layer [29]. Finally, the tertiary layer acts to control the power flow management of the distributed generators [30], [31]. Figure 1 presents a general scheme of a three-layer hierarchical control.

As previously mentioned, the hierarchical layers divide the strategies according to the operational dynamics, so it is possible to decouple the design and analysis of each of its layers [32]. The interest of this work is to analyze the operation of the primary and secondary layers. The findings presented can be generalized to operate in conjunction with different tertiary layer strategies.

The use of communication systems is also fundamental in the way hierarchical control operates in isolated microgrids. The strategies can be classified into three configurations: centralized, distributed, or without communications [33]. In the first category, the strategies are governed by a microgrid central controller (MGCC). The DG controllers send measured signals to the MGCC, which receives and manages the information. The MGCC performs calculations and sends back control signals to the DGs periodically. In the second category, the strategies operate distributed, so the MGCC is not required since the control system is designed to rely on the data interchange between the DGs. These categories have in common the use of communication systems for their operation, making them vulnerable to communication issues such as link failures, data transmission delays, and noise disturbances, among others [14], [15].

Although in the second case, reliability improves since it does not depend on the correct functioning of a single equipment (the MGCC), the complexity of installation can increase when numerous DGs are connected. For these reasons, the third category has emerged, in which decentralized control strategies that do not use communication systems are proposed [16]. These types of strategies eliminate the risk of operational failures due to communication issues, although these do not avoid the need to implement a MGCC for other functions such as the DGs coordination during the black start process. Since fewer control layers and operational actions rely on the MGCC performance, the MG will be more reliable [34].

The droop control can be defining in equation 1 for frequency regulation as follows

where the frequency ω rely on the active power P, ω ₀ and m are frequency reference and droop control gain, respectively. Active power P corresponds to the instantaneous active power p filtered using a low-pass filter with a cutoff frequency ω_c such as equation 2 to achieve high power-quality injection

where, P(s) and p(s) are Laplace transform of P and p, respectively.

This general definition of droop control is decentralized as it operates exclusively using the local measurements (p and ω). The error in the operating frequency concerning the reference o₀ can be recovered by the secondary layer, adding an extra term δ as follows

According to how the delta term is calculated, the secondary layer is classified.

The decentralized proposal presented in [20] defines δ as

where k_i is a constant parameter and /c(t) is a control parameter driven by a time protocol (sgn corresponds to the sign function or signum function). This time-domain expression can be analyzed in the frequency domain as a switching control with two operation modes: a filtered proportional controller and an integral controller, as follows:

being C the last value calculated by the filter just before k(t) = 0. Figure 2 presents a scheme of the operation of this proposal. Notice that the value of k(t) controls the transition between modes and affects the cutoff frequency of the low-pass filter. Thus, the time protocol can drive the value of k(t) to obtain the best features of both the filtered proportional controller and the integral controller. This proposal is characterized by a fast speed of response and a good frequency recovery.

Figure 2 Source: elaborated by the authors.

Figure 3 presents the time protocol for variable k(t). This time-protocol is launched when an "event" is detected. According to [20], an "event" is a change in the power delivered or in the frequency of the MG caused by the connection and disconnection of loads and new DGs. Thus, it is possible to implement different approaches for the detection, such as a send-on-delta strategy based on a definition of thresholds for variables such as frequency or active power rate variations [35]. The time protocol is composed of three operation zones: a first operation zone that starts when the event is detected at t_e, in which k(t) has a constant value of k_max during a duration of Δ_et. Then, a second operation zone in which k(t) changes in a linear ramp from k_max to 0 during Δ_ramp, according to equation 5 these operation zones correspond to the filtered proportional controller. Finally, a third operation zone after t = t_ramp, the switched control maintaining the last value calculated (C in equation 5).

Figure 3

This strategy was designed to operate on balanced microgrids. However, as will be verified later, the presence of unbalanced loads can lead to problems with the power-sharing performance. For this reason, an additional control block is proposed.

Figure 4 shows a general scheme of a grid-forming inverter. The DC voltage source presents the DC stage composed of a distributed power source connected through a converter and a DC-link capacitor. At the output of the IGBT power inverter, an LC filter (L_ín, C) is connected to reduce the high-frequency harmonics [36]. Then, the elements L_T and R_T represent a coupling transformer to connect the inverter to the point of coupling.

Figure 4 Source: elaborated by the authors.

The sensed signals are i, v, i₀ that correspond to the inverter output current, the voltage at the capacitor of the LC filter and the current injected to the coupling transformer, respectively. These signals are transformed to the aß framework through the Clarke transformation, and their fundamental positive and negative sequence components are extracted through a double second-order generalized integrator (DSOGI) [37], [38].

The fundamental positive sequence components, v⁺ _1αß and i⁺ _o,1αß, are used for the primary layer block (droop control) and the instantaneous active power p is calculated according to [23] as follows:

The secondary layer based on the switched control is highlighted in green in Figure 4. It receives the frequency ω and calculates the compensation parameter δ for the primary layer to mitigate the frequency error.

As previously indicated, the proposed strategy is inspired and adapted from the work presented in [23]. The proposed control corresponds to a Q^- - Z block, which is based on the calculation of a virtual output impedance as a function of the negative reactive power. First, the negative reactive power Q^- is adapted by [23] using αβ components instead of components.

This reactive power term is used to calculate an output impedance Z using the expression:

where Z₀, k^- and Q_o ^- are an initial output impedance, a proportional negative control coefficient, and an initial negative reactive power, respectively. These parameters can be set to improve the operation of the proposed strategy. Then, the term Z is operated with the negative current i_o1 ^-, as

This v^- _αβ,ref signal is applied to the primary control of the hierarchical control structure. The block of the proposed control is highlighted in red in Figure 4.

First, the simulation was performed using only the primary control layer. The same value of the droop coefficient m was implemented for all the DGs considering converters with equal nominal powers. The results for the instantaneous three-phase active powers and frequencies are presented in Figure 7.

In these figures, it is possible to clearly observe the effect of the droop method. The primary layer performs fast and accurate power-sharing between the DGs, despite the unbalanced load. However, as previously discussed, the droop characteristic produces a deviation in the frequency. From t = 0[s] to t = 15[s], when only DGi is operating, the frequency presents the greatest deviation because this inverter must deliver all the power demanded. Once the other DGs are connected, the deviation is reduced since the power-sharing is performed.

Next, the simulation was performed using the secondary switched control presented in [20]. The results for the instantaneous three-phase active powers and frequencies are presented in Figure 8.

Figure 7 Source: elaborated by the authors.

Figure 8 Source: elaborated by the authors.

The operation of this control strategy requires detection of "events", which in this case are the connection of the DGs at t = 0[s], t = 15[s] and t = 45[s], and the connection of the unbalanced local load Z _L at t = 30[s].

Since before t = 30[s] the microgrid operates with balanced loads, it is possible to appreciate the proper functioning of the strategy. During the first operation zone that starts when each event is detected (see figure 3), an accurate power-sharing is done during 5[s]. However, an appreciable error in the frequency recovery can be noted. Next, the parameter k(t) changes in a linear ramp during 5[s], compensating for the frequency error and showing a good dynamic response. Finally, once the frequency deviation is eliminated, the control switched maintaining the last value calculated until the next event is detected. However, despite the good results of the control strategy when the microgrid is balanced, from t = 30[s], once the unbalanced local load Z _L is connected, an error in the steady-state power-sharing is appreciated. Between 40[s] and 45[s], the difference between the active power delivered by DG ₁ and DG ₂ is around 50 [W].

Furthermore, between 55[s] and 60[s], the difference between the active power delivered by D G ₁ and D G ₂ , and DG ₁ and DG ₃ are around 55[W] and 40 [W], respectively. This is because the switched secondary control strategy fails to act over the negative sequence component of the generated powers.

Finally, the simulation was performed using the proposed secondary control layer for unbalanced loads. The results for the instantaneous three-phase active powers and frequencies are presented in Figure 9.

Figure 9 Source: elaborated by the authors.

These results allow verifying that the proposed control mitigates the power-sharing error observed in the previous simulation. Thus, with the added control block, it is possible to maintain the performance of the switched control (good dynamic response and frequency restoration in steady state), without being affected by the impact on the negative sequence generated by the operation of unbalanced loads.