A number of general analyses and solution techniques have been developed for the current problem of secondary voltage frequency regulation in multi-parallel inverter microgrids. In this section, some of these key technologies are introduced and analyzed in terms of their implementation methods and principles. Development trends of the methods and technologies are also analyzed.

Virtual impedance is a classical method used to improve the stability and reactive power-balancing accuracy of multiple parallel inverter microgrid systems [68]. A schematic diagram of the virtual impedance method is shown in Figure 1.

R_v and L_v are the resistive and inductive parts of the inverter virtual impedance. U_tm is the output port voltage, and I_o is the inverter output current. The principle of the virtual impedance method is that the output port voltage of the inverter is changed to U_tm by virtual impedance loop calculations, as if an equivalent impedance of R_v and L_v is connected in series between the inverter output port and the original output port behind the droop-control loop.

The dq-axis virtual impedance expressions are as follows:

Since the inverter control inner loop is the decoupled dq-axis PI controller, Equation (9) is commonly used for the implementation of the virtual impedance method. Thus, based on the inner current-voltage control loop illustrated, the virtual impedance control technique can be achieved by changing the reference inputs of the voltage loop from the original u_c,d^* and 0 to u_tm,dq.

The key problem with the implementation of Equation (9) is that the differential signal for the output current in it is difficult to realize. Differential calculations will lead to a significant amplification of the signal and numerical noise and should generally be avoided in inverter control loops. There are two main ways to handle the differential calculation issue. One solution is to approximate the differential link as a high-pass filter. Then, the control method in Equation (9) becomes:

The virtual impedance method is defined as transient virtual impedance in [69]. The other solution for differential calculations ignores the dq-axis current differentiation, and the virtual impedance form becomes:

A significant effect of virtual impedance control is that it can improve the accuracy of droop inverter reactive power distribution. Considering Equation (8), when the line impedance of the system is proportional to the droop coefficient, that becomes:

Ignoring the voltage drop due to the reactive droop loop, the Q-U droop relationship will be restored according to Equation (8). Generally, it is difficult to achieve the condition that the line impedance is proportional to the inverter droop coefficient. However, with the virtual impedance control technique, the relationship in Equation (12) can be restored with the virtual impedance technique. In addition, the resistive part of the virtual impedance can enhance the small-signal stability of the system [70]. There has been a considerable amount of research on virtual impedance methods and their applications. Some of these typical studies and their corresponding abstracts are reviewed in Table 1 [37,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89].

The consensus algorithm is derived from graph theory. Consider a fully connected undirected graph Ξ containing M nodes inside. G is the set of all edges formed in the graph Ξ. Y_ij denotes the edges connecting node i and node j.

A is defined as the M × M adjacency matrix with the elements a_ij defined as follows:

Then, define D as the connectivity matrix whose elements are defined as follows:

Finally, define L as the Laplace matrix, and it becomes L = D − A. The matrix L is a symmetric matrix that has 0 eigenvalues, and its corresponding eigenvector is an all-one vector. Furthermore, according to Green’s disc theorem [38], it is known that the matrix L is a semi-positive definite matrix, which becomes the reason that the system with the consensus algorithm is stable. A brief proof of the stability of the consensus algorithm is presented in the Appendix A.

The widely adopted leader–follower type of consensus algorithm is constituted as follows:

In Equation (6), x_i is the state variable, and u_i is the control input. b_i is the pin coefficient. b_i = 1 if the i-th node is the leader node, and b_i = 0 otherwise. It can be proved [38] that the dynamic of (6) eventually converges to the reference value x^ref for all states x_i. If the state variables in (6) are the frequency or voltage variables of a multiple parallel inverter system, the frequency and voltage of the inverters in the system will be regulated to the target value.

Recently, the consensus algorithm has been widely applied to solve the secondary-frequency and voltage-regulation problem in microgrids. However, the consensus algorithm will bring immense communication pressure to the inverter. In order to reduce the communication pressure caused by the consensus algorithm, the consensus algorithm with an event-triggering mechanism improvement is gaining more and more attention [90].

The event-triggered mechanism is similar to the idea of variable structure control [91]. The system control state updates and broadcasts its state when the system reaches a certain threshold away from the stable equilibrium point. The controller remains silent when the system is within the state threshold or the state reaches the convergence phase plane, at which point the system will spontaneously converge to the stable state even if the control state is not updated. Figure 2 shows the regulation pattern of the system with the event-triggered mechanism.

Research on the application of event-triggered mechanisms to improve the consensus algorithm in order to optimize the frequency and voltage regulation process of multi-parallel inverter microgrids is presented in [92,93,94,95,96,97,98]. To summarize the existing designs and applications of event-triggered-based consensus algorithms, the following highlights can be mentioned regarding the design of event-triggered mechanisms.

The term “event” in the event-trigger mechanism refers to the event that the event function f_j is greater than 0. This means that when the event-trigger function satisfies f_j(t) ≥ 0, the inverters broadcast their latest output signal to the connected inverters connected on the communication network. Further, when the event-trigger function satisfies f_j(t) < 0, the individual inverters remain silent, while the consensus algorithm of each inverter uses the last signal sent by the inverter communicating with it. There is a certain pattern to the design of event-trigger functions for event-triggered mechanisms, which are often in the following form:

M(θ) is a specific function consisting of various types of coefficients related to communication or network structure parameters. e_j(t) is the error function consisting of the actual value of the state variable and the consensus state variable, and z_j(t) is the consensus algorithm function. For example, the event-trigger function in [92,96] are both in this form.

The Zeno phenomenon refers to infinite triggering times in a short period of time [95]. No Zeno phenomenon is an important criterion for the design of event-triggered mechanisms. In general, the method used to prove that no Zeno phenomenon will occur is to show that there is a lower bound for the event-trigger time from the designed method.

Numerical chattering can lead to frequent meaningless event triggers after the consensus algorithm has reached a steady state. Therefore, to further reduce the event-trigger frequency, the absolute or squared value of the error function e_j(t) or the consensus algorithm function z_j(t) can be used as an indicator to set the corresponding threshold for an anti-chattering design. However, the anti-chattering threshold should not be set too large, or it will affect the stability of the event-triggering consensus algorithm.

In [92,93,94,95,96,97,98], the event-triggered consensus algorithm is adopted to design the secondary-frequency and voltage-regulation method, but the consensus signals are selected as the frequency and voltage signals themselves. According to the previous discussion of droop-based control, an important requirement in the design of the secondary-frequency and voltage regulation link based on droop control is that the secondary-frequency and voltage-regulation component of each inverter should be of the same value; otherwise, the droop characteristics will not be maintained. Therefore, in order for the droop characteristics to remain unchanged, the existing event-triggered consensus algorithms for secondary-voltage and frequency regulation adopt additional components to maintain the invariance of the droop relationship among inverters. For example, in [92,95,96], the power differential signal is introduced to compensate for the original consensus algorithm to maintain the droop relationship. In [97], an additional active power droop consensus component u_Pi is added to the consensus algorithm, and the whole method becomes complicated.

At present, the theory on and analysis of consensus algorithms with event-triggered mechanisms are not yet mature. The impact of the event-triggered mechanism on the small signal and transient stability of microgrid systems has not yet been fully studied.

As mentioned before, non-ideal operating conditions such as disturbances, harmonics, and imbalances are common in the operational control of multi-parallel microgrids. Therefore, control methods for frequency and voltage-regulation control under various types of disturbances and uncertainties are widely used. Among them, active disturbance rejection control (ADRC) and robust control are effective control methods for various types of disturbances and uncertainties [99].

ADRC is fundamentally different from robust control. ADRC emphasizes the observation of and compensation for these perturbations, and the control design procedure concentrates on the design of the perturbation observer. Robust control emphasizes the minimization of the effect of model uncertainty on the stability of the system by designing a stabilizing controller. Therefore, the design procedure concentrates on optimizing a specific robust stability criterion. In addition, the source of perturbation that is the focus of concentration by ADRC is generally external perturbation, whereas the robust control is generally targeted at model uncertainty.

The key design procedure is to design the extended state observer (ESO). (8) is known as the extended state observer (ESO). β₁ and β₂ are known as the observer gains.

The advantage of the ADRC control method is its ability to effectively suppress external disturbances, enabling the system to recover to the steady-state equilibrium more quickly after being disturbed. The ADRC control method has been applied in some research to design control methods for microgrid multi-parallel inverter systems with enhanced system immunity [100,101]. Ref. [100] used the ADRC control method to enhance the dynamic immunity of a frequency/voltage secondary regulation scheme based on a consistency algorithm. In [101], the ADRC method is used to enhance the dynamic performance of microgrid inverters.

The key to design a robust controller is to model the uncertainty of the system. System uncertainty can be divided into additive uncertainty and multiplicative uncertainty. Define z as the uncertainty model output and w as the uncertainty model input. According to the transformation between the state space and the transfer function, the matrix of the transfer function from w to z can be obtained as follows:

The H_∞ control optimality problem can then be formulated as finding the control gain matrix K such that the H_∞ norm of the transfer function H_zw(s) is minimized:

Furthermore, the H_∞ control suboptimal problem is formulated as finding a control gain matrix K that satisfies the condition that the H_∞ norm of H_zw(s) is smaller than the given constant γ:

The basic principle of H_∞ control in robust control is presented above. A detailed frequency domain H_∞ controller design flow is given in [102] for a typical photovoltaic DCDC converter. Summarizing the design in [102], the design of the frequency domain H_∞ controller is concluded in the following flow.

(a) Transform the uncertainty terms in the control model into additive uncertainty form or multiplicative uncertainty form.

(b) Extract and separate the uncertainty terms by linear fractional transformation (LFT). Then, select the suitable weight function W and integrate it into the model G, resulting in a nominal model/uncertainty term/control matrix corresponding to the G-Δ-K form.

(c) The H_∞ optimal or sub-optimal problem is solved by μ synthesis algorithm to obtain the required control gain matrix K.

There have been some studies and applications of H_∞ control for the problem of the secondary regulation of frequency and voltage in a non-ideal microgrid [103,104,105,106,107]. In [103], the full flow of the design and stability analysis of the H_∞ controller for grid-connected inverters is presented. The H_∞ problem is solved by the μ synthesis algorithm. In [104], an inner-loop sliding-mode control and an outer-loop hybrid H₂/H_∞ control was adopted to design a grid-connected inverter with strong robustness, and the linear matrix inequality (LMI) algorithm was used to solve the H_∞ problem. In [105], an H_∞ controller is applied to a frequency quadratic regulation control loop. The H_∞ problem is solved by the μ synthesis algorithm. In [106], the H_∞ controller was used to design the control inner loop of a microgrid inverter for islanded operation, and the LMI algorithm was used to solve the H_∞ problem. In [107], H_∞ controllers are designed for improving the small-signal stability of microgrid inverters against communication delays and non-linear load conditions.

For the application of robust H_∞ control methods in the field of frequency/voltage secondary regulation of microgrid inverters, the number of applications and the amount of research are still relatively small, and a unified scheme has not yet been developed. More in-depth and standardized work is desired in the future.