2.1. Reactive Power Sharing Error

2.2. Adaptive Virtual Impedance Droop Control (AVIDC)

The adaptive Z_V−-based controllers can be implemented without communication network [14, 62, 88–93, 106–111] or with communication infrastructure [4, 8, 13, 15, 16, 27, 41, 43, 44, 55, 61, 66, 94–100, 112]. In [101], a virtual impedance scheme is presented to control the virtual power supply, which consists of two parts: a negative resistance against line resistance along with P/Q power decoupling and a negative inductance to calculate accurate reactive power sharing by virtual voltage drop. In [103], an adaptive control technique is proposed for P/Q power sharing in a hybrid AC/DC MG equipped with a solar panel module/battery to support maximum reactive load demand. In [104], a reshaped MG is studied by an AVIDC approach, which is applied to grid-forming VSCs to rearrange units in the form of ideal AC voltage sources [103, 104].

3.1. Centralized Structure

Centralized control is known as the most traditional, most accessible, and most common control structure of power systems, which has highly accurate results due to the maximum reliance on the central controller. However, the centralized structure due to its reliance on both LBC and high-bandwidth communication (HBC) has the lowest security and reliability as it relies on the MG central controller (MGCC) as the mastermind of control. A centralized controller usually has a slower response time than a decentralized one like a droop controller. In such a controller, all the information related to the bus voltage, the frequency of the units, the P/Q output power, weather forecasting, etc., should be collected and sent to MGCC [61, 124, 125]. The MGCC is responsible for transferring and collecting information by an easy decision-making process [103]. This task is done by receiving commands from supervisory controllers, SCADA, and advanced data processing software such as advanced distribution automation (ADA) for controlling and monitoring purposes [126, 127]. The MGCC updates droop control parameters and gains every time interval (several minutes), quickly. After data aggregation from local controllers (LCs), the control pulses are synchronized using a phase-locked loop (PLL), and then, the current signal and V/f reference parameters are reflected in VSCs through HBC graphs [10, 41]. Common power regulators installed in each unit provide a global observation for precise tracking of the optimal set points and achieving cooperation between electronic-interface DERs [46]. Another advantage is interaction with demand-side management (DSM), which is based on end-user responses to balance supply and demand. In [14, 41, 61, 103, 128], the centralized-based AVIDCs are discussed according to arbitrary X_V/R_V ratios to control Q − V, while these are not required to use the communication between units; thus, it causes a decrease in delay. However, in [14], the IRPS is not compensated, accurately.

In the centralized structure, unlike the local one, the recalling information is transferred to the central controller, and conversely, the commands from the controller are transferred to sources through a communication link directly. As an example, a DER with the least changes is defined as the reference unit, and the output values of other DERs are compared to it to update by MGCC in real time. The updated values are then sent to all units to estimate the virtual impedance. Instead, in the centralized structure, the plug-and-play feature cannot be obtained because when a new source is added to MG, it is necessary to readjust the MGCC [14].

3.2. Decentralized Structure

This pattern is based on the allocation of V and f measurements in IMG upon predefined algorithms to solve the IRPS [133, 134], harmonic voltage distortion at the PCC, harmonic current/power sharing [135, 136], and V − f regulation just in a single-level structure. The decentralized control usually is applied to configure an MG with a large number of devices compared with the centralized structure [11, 18, 27, 29, 62, 70, 81, 90, 94, 102, 112, 113, 117, 118, 122, 124, 125, 137–146]. The decentralized control often is synonymous with the droop method, which is applicable just for off-grid mode. As the decentralized structure is inherently independent of the communication network, the advantages of the droop method such as reliability [10, 125], inviolability against delays, high-speed performance, cost-effective implementation, and the plug-and-play mechanism are often similar to the decentralized control features [70, 73]. However, for the same reason, it suffers from the IRPS, lack of flexibility, and diversity of the control system [7, 10, 11, 53, 54, 70, 86, 102, 113–120, 124, 125, 147]. These problems become more catastrophic when nonlinear load characteristics, harmonic power components, and voltage unbalance at the PCC are involved in the MG. [40, 72, 136]. In [29], a decentralized impedance-based droop method has been presented to reduce power loss. Nevertheless, the IRPS has not been analyzed, directly. In [142], a fully decentralized scheme is intended by using dead time to equalize the power coefficients in the LCs so that it leads to regulating the voltage drop affected by IRPS.

Some scholars believe that MGCC-based secondary control is another form of decentralized control, which is implemented according to a hierarchical structure [26, 53, 76, 82, 83, 149]. A lot of harmonic power sharing techniques [10, 13–20, 125], optimization issues (refer to Section 6), intelligent algorithms [94], etc., are often included in this category. For example, in [94], a fast servo system in decentralized control is presented to optimize voltage and the reactive power sharing by a fuzzy particle swarm method in off-grid mode.

The decentralized control structure is categorized into the following two groups.

3.3. Hierarchical Structure

The hierarchical pattern is usually identified by three control layers, which are organized according to ANSI/ISA-95 (International Automation Society). It is a management standard to recognize applicability between different organizations and control systems [53, 114, 143, 146, 150–154]. The important point during the hierarchical control process is to adopt a control approach with discrete-time scales to reduce delay and less complexity for data exchange. According to Figure 4, by adding each of the control layers, MG requires a communication graph implemented through a consensus protocol to exchange information (refer to subsection 3.4.2). Therefore, the hierarchical structure suffers from delay and sluggishness despite being accurate. Sometimes, the authors merge the secondary and tertiary control levels with the objective of reducing operating time [10, 149, 155, 156].

Features of the mentioned three levels are summarized as follows.

3.4. Distributed Control

The idea of using distributed control in the form of SG was proposed in the mid-2000s for the first time [184]. Generally, all the control approaches of MGs are based on a distributed control paradigm until it is proven against this principle.

The hierarchical structure is an advanced configuration of decentralized control, so it can be expected that the hierarchical structure inherits the advantages of decentralized and even centralized control structures. Meanwhile, it has left behind a large part of their challenges [6, 23, 27, 38, 50, 53, 58, 64, 72, 73, 96–98, 154, 160, 163, 185–189]. In some reports, the authors believe that the distributed, decentralized, and hierarchical control are not different from each other [6, 23, 27, 30, 38, 39, 50, 58, 64, 72, 73, 97, 98, 134, 160, 163, 183, 185–193]. For example, in [190], the authors have preferred to introduce the two levels of control in a decentralized structure instead of defining it in an advanced hierarchical one. That is, in the fully distributed control, each controller requires just information exchange with its neighbors according to the bidirectional communication graphs without relying on a central controller [73]. However, sometimes unidirectional graph is implemented for the secondary controllers in distributed control form [185]. In [67], a robust nonlinear distributed approach is proposed to maintain the system stability and realize the P/Q power sharing with fast dynamic performance. In [111], an AVIDC is studied that synchronizes the output voltage of the DERs based on the difference between the output active power of each LC and the average active power signal to other LCs. In [73], a fully distributed control approach is presented that defines three control modules: (1) the voltage regulator, (2) the reactive power regulator, and (3) the (P − ω) regulator. The second module adjusts the (Q − V) droop coefficients to reduce the IRPS by considering the effect of the mismatched output impedance of units. Unlike most classic controllers, no frequency measurement is used in this AVIDC.

The examples mentioned are just a definition of conventional distributed control or hierarchical structure. Therefore, there are still fundamental problems such as delays, huge costs, complexity of control algorithms, and lack of trade-off between voltage regulation and reactive power sharing. Thus, advanced distributed coordination control is required.

3.4.1. Distributed Coordination Control Based on MAS

The most complex platform for SGs is the MAS-based distributed coordination control so that each DER and its internal LC act in the role of an intelligent agent [35, 57, 86, 105, 153, 186, 187, 194–196]. Global synchronization is the main control objective of agents preprogrammed in the artificial intelligence language. Agents are semiautonomous facts that exist in a software environment. Each agent can converge its conspecifics in a peer-to-peer model relative to itself through communication graphs [197]. It causes them to influence them in other environments and changes that environment according to a specific control motivation. The environment can be defined by tracking the conditions in which agents have the least penetration to change them. As agents are processed in an independent environment, these are self-sufficient factors to observe their performance [71]. Nevertheless, agents cannot be independent of their programming operator; therefore, these have just a significant degree of independence, which must be considered for a system and target [117]. In other words, each agent has a common target with other agents in a complex control system as an SG [197].

The agents can be divided into three main groups as follows:

• 1.

  Active agents (leader agents): It includes agents that are pinned in the control computation path as a dispatchable reference agent [39, 57, 99, 196].

• 2.

  Reactive agents (passive/follower agents): It includes agents that are considered a nondispatchable obedient agent in the control computation path [39, 57, 99, 196].

• 3.

  Complex agents (combination of 1 and 2) [39].

The leader–follower agents have a self-sufficient performance inspired by many control methods, which can control SG by time-varying communication infrastructure. The MAS-based distributed coordination control acts like a master–slave system for proper current sharing except that the master–slave control relies on the HBC link and receives commands from the central controller [26, 53, 76, 82, 83, 125, 198]. The MAS algorithms provide solutions for sending, receiving, analyzing, processing, accepting, and rejecting the information of other agents in a coordinated configuration [39]. In some papers, the MAS is introduced as a hierarchical or decentralized control frame [35, 57, 86, 96, 105, 113, 153, 186, 187, 195–199]. As the VSCs are inherently passive in a classic hierarchical structure, required to receive commands from MGCC, the MGCC or central controller is removed from MAS [40, 72]. Alternatively, the MGCC encounters difficulty in covering the whole information of the system in comparison with MAS [10, 21, 23, 86, 187, 195].

From the point of view of the MG architecture, the agents can be divided into two categories [1, 116, 186] as follows:

• 1.

  Resource agents: These include load agents, controllable load agents (CLAs), DER agents, and ESS agents at the primary layer for monitoring states of elements as setting operating points of LCs, state of charge of ESS, and dispatchable capacity of CLA.

• 2.

  MG agents: It includes the MG operation agent (MOA) for managing the MG/MG cluster along with processing the optimal generation of dispatchable DERs and the MG market agent (MMA) for marketing the operation of the MG. MG cluster consists of several MGs converged in a coordinated pattern so that the MMA is responsible for negotiating with the MMAs of other MGs to purchase/sell power from/to them. The MG cluster must have a relationship with the distribution network agent (DNA) to comprehend the constraints of the energy market (see Figure 5).

It is worth mentioning that the same three levels of the classical hierarchical structure are maintained in the MAS except that MGCC is removed. Each level has different dynamics and time scales; also, each LC is a hybrid agent, which has two reactive and deliberative layers [39, 67] as follows:

• 1.

  The reactive layer is often responsible for the voltage synchronization of buses through information exchange between agents. The process provides precision, recognition, and action models.

• 2.

  The deliberative layer is the starting point for optimization issues such as self-sufficiency, realization of vision, and potential action models.

Generally, the most important challenges of MAS-based distributed coordination control are as follows:

• •

  The MAS has two types of functional nature: physical or virtual character. In terms of physical nature, an agent can be controlled in the role of a DER. However, virtual agents participate in the energy market, and therefore, these sell/reserve the measured surplus power [117].

• •

  The presence of MAS in the electricity market, the complex relationship with DNO and MO in the third layer, and compliance with the local facilities on the customer side are the management challenges of MAS for supply and demand [10, 21, 86, 195].

• •

  The programming-based complexity and the lack of professional knowledge of human operators cannot be ignored for the implementation protocols in MAS.

• •

  The trade-off between the improvement of IRPS and V − f regulation is an important challenge for IMGs controlled by MAS.

• •

  Dependency on the communication links that rely on the intervention of the global Internet to send/receive data to/from agent consensus by peer-to-peer structure leads to complicated problems such as the modeling complexity, mathematical computations, stochastic delay, costs, reliability in confronting cyberattacks, and the plug-and-play mechanism [84, 116, 117].

• •

  The MAS suffers from stochastic delay because agents must use bidirectional communication graphs to exchange data with each other similar to the hierarchical structure.

3.4.2. Consensus Protocol

The MAS is the most developed distributed coordination control strategy according to the consensus algorithm [40, 161]. MASs are inherently adapted to the environment conditions because of the implementation by the node-graph (edge) protocol in each MG synchronization cycle.

Two categories of adaptive consensus protocols reign in the secondary level of MASs [44–46]:

• 1.

  An edge-based adaptive consensus protocol that couples two nodes by a pair weight graph as a peer-to-peer network.

• 2.

  A node-based adaptive protocol in which each node/agent is a database for the convergence of bidirectional LBC graphs that carry information about the environment (IMG), while time-varying weights manage the dynamic response of the MG.

The decisions that are made by the interaction between agents at the secondary level are targets that are obtained from the planning of agents at the primary level. The secondary control function is switched by a triggering signal to start or terminate in a close-loop path [86]. Then, targets are emitted throughout the SG through an intertwined communication network called a peer-to-peer protocol, which can be defined by graph theory. A coordination consensus algorithm is a fundamental platform of a multilevel system, which inherits graph theory in the form of LBC links to optimize the convergence of agents as peer-to-peer networks [51, 52, 116]. The graph theory considers each bidirectional LBC link as an edge or graph as well as considering each agent/LC/VSC/DER as a node that configures a tree to achieve the MAS-based distributed coordination control [10, 70, 113, 116–118, 124, 200]. Because LBC links have time-varying dynamic behavior, the graphs cannot be inherently balanced [23]. There is a coordinated relationship between the graphs within every time interval scheduled in a finite range [201]. Without expressing complicated equations, each node can be modeled as follows:

$$ ()

where V_G represents the set of nodes (agents) and E_G is the set of edges (LBC links) as follows:

$$ ()

Now, a question may be raised due to the unreliability of bidirectional LBC links under the threat of cyberattacks: How are interrupted graphs identified in the peer-to-peer network? The weights are the most basic concept in the MAS control, which can express the outage mechanism, failures, and cyberevents by a diagonal matrix as A_G. The adjacent matrix A_G by the set of arrays a_ij represents the concept of weights for each graph as follows:

$$ ()

If the a_ij = 0, it means the disconnection of the communication link/graph in the MAS. Also, if a_ij = 1, the connection between agents is established, given that a_ij ≥ 0. The index i is related to ith controllable agent/DER, which is a reference agent for calculation. Also, the index j is for jth adjacent agent/DER. These weights form the arrays of $$ for in-degree matrix D_G = diag ∈ {d_I} as a diagonal matrix. The adjacent matrix A_G can be rewritten as follows:

$$ ()

As the consensus protocol does not rely on comprehensive communication between units, it is just run by weighted average. In the IMGs, it can result in power average control along with a plug-and-play mechanism [16, 27, 94, 97, 105]. According to Figure 6(a), assume that VSC₂/agent₂ is a leader and the other DERs are follower agents. If a_i = a₂ is ith weight, a₂ is in connection with the other three weights as a₂₁, a₂₃, and a₂₄ at the second row of the A_G. In the same way, this condition is maintained for each agent pinned as a controllable reference agent in each synchronization cycle, for example, if sending information from agent₁ to agent₄ is not available or is subjected to cyberattack (see Figures 6(a) and 6(b)); therefore, the adjacent matrix A_G can be rewritten as follows:

$$ ()

Finally, a Laplace matrix can be defined to obtain a spinning graph called L, where it addresses the optimization of the MAS [113] as follows:

$$ ()

where D_G eigenvalues determine the global dynamics.

Now, assume that the ith DER is a reference agent (e.g., DER₂ in Figure 6(a)) and j index is related to the neighbor DERs/agents. Thus, the reactive power sharing in IMG controlled by MAS is as follows:

$$ ()

where u_qi is the input signal to the reactive power controller as follows:

$$ ()

where k_q is a proportional gain. The handoff of information between the agents is done as a coefficient of the sum of the to and fro signals with a specified tag. For example, in IMGs, the IRPS error δ_qi entered into the ith unit relative to the jth unit is as follows:

$$ ()

where n_i and n_j are the droop gains and Q_i and  Q_j are measured reactive powers of ith and jth agents, respectively [22]. Accordingly, the system must be able to continue the P/Q power sharing process with a fast dynamic response to eliminate circulating.

Almost all the graph theory-based algorithms ensure the selected variable tends to the nominal value in a steady state [1, 105, 196]. In the case of the MG cluster, Equation (11) can be generalized for P/Q power sharing between n-MG, n = i, …j as follows:

$$ ()

$$ ()

where $$ represents active and reactive references indicated by the tertiary level, which are optimized based on constraints of the energy market. Also, $$ and $$ are the average angular frequency error signal and voltage of the ith unit relative to the jth unit, respectively.

The consensus-based techniques can guarantee voltage deviation for the management of IMGs controlled by hierarchical and MAS structures, which are under threat from the IRPS [10, 21, 36, 40, 43, 70, 72, 97, 105, 113, 116–118, 124, 200]. Generally, the classification of consensus-based structures that can realize self-organizing control to improve or tackle SG problems is as follows: consensus with constraints [198, 201], event-triggered consensus [20–22, 24, 37, 58, 60, 71, 79, 109, 132, 134, 151, 180, 200, 202, 203], finite-time consensus [35, 46, 50, 52, 65, 183, 186, 204, 205], linear consensus protocol [206], heterogeneous consensus [43, 45, 127, 203, 207, 208], nonlinear consensus [23, 38, 45, 162], distributed averaging control [25, 161, 209], pinning-based distributed control [4, 8, 25, 86, 98, 106, 108, 150, 153, 161, 192, 199, 210, 211], and consensus programming-based control [23, 32, 48, 50, 55, 74, 106, 116, 127, 163, 198].

4.1. Pinning Control Algorithm

When the MG is disconnected from the bulk system, control commands are not received from the supervisory center; therefore, the new commands are received from the side of the agents. The pinning algorithm can pin a selected agent in the role of leader in each synchronization cycle of a MAS to a flagged environment. To select a set of pinned-leader and unpinned-follower agents, the distance and degree indexes are defined for the connection of the leaders relative to other MG devices [153, 199]. The process is defined in the last or third control layer to observe the output values [153, 192]. At the same time, the highest degree of authority of unpinned/follower agents is implemented for the last control layer to resolve V and f drop at the PCC. It means that a voltage observer receives the output voltage of all the DERs and then calculates the average voltage value at the primary level. Finally, the P/Q power sharing will be achieved based on pinned candidates at the secondary level [25, 153]. Thus, the dynamic MAS-based consensus averaging control must be pinned to at least one of the agents placed in the outermost control layer (second or third layers) [108, 150]. At the boundaries of the lower and upper layers, reference signals are extracted by the algebraic equations. These bounds can define a suboptimal algorithm with polynomial complexity. It flags nodes that are selected to pin the optimal network agents, efficiently [25, 199, 210]. It is worth mentioning that the operation of unpinned agents precedes the reaction of protective relays in the IMG [161, 199].

In several papers, the pinning algorithm can guarantee the performance of MASs to improve the IRPS [25, 86, 153, 196] and voltage regulation at the PCC [86, 153, 192, 210]. By some intelligent techniques such as the noroadaptive approximation technique, the fully distributed self-adjusting is asymptotically achieved. This situation holds even though the upper bounds of the nonzero inputs of the leaders and also the neural network approximation error are ambiguous for MAS [106]. In [199], the authors have proposed two adaptive algorithms for selecting pinned nodes(s), while it uses the degree and distance index of the selected leader(s) to other IMG segments according to [211]. Besides, a small part of the pinned agents is just capable of nonadaptive performance by simple feedback control. In [196], a droop control scheme is proposed for P/Q power sharing at the primary level of off-grid mode, while attaching a secondary level to the pinning algorithm implemented at the primary level completes the accuracy of trade-off (Q − V). In this case, the voltage source agents are pinned as virtual leaders. It means that the load agents are selected as a reactive and passive mechanism that is not pinned. In [86], a two-layer MG cluster is modeled, which generates current and voltage references based on P/Q power sharing errors between all the DERs. The pinned agents share the reference values with their neighbors at the secondary and primary levels for tuning voltage and frequency to nominal values. In [210], the ESS is considered as the leader agent by the algorithm-based pinning in the IMG.

Analyzing the above cases shows the most important advantages of the pinning-based MAS are to decrease controller/agent intervention and to make mathematical modeling more efficient. The most important form of the pinning control is relying on the NP-hard algorithm. Although NP-hard can solve the most complex computational problems with nonlinear binary programming, it causes the complexity of calculations that are classified in nondeterministic polynomial times.

4.2. Distributed Averaging Consensus Algorithm

In an MG with k paralleled DERs, k = i, … j that is connected at the common current bus, LCs are equipped with a control circuit, which is not individual, necessarily (see Figure 9(b)). The added current control loop tracks the reference current provided by the common current bus. Moreover, a voltage observer located in each LC tracks the bus voltage, voltage drop at the PCC (in single-bus MGs), and output voltage of jth neighbors in each calculation cycle. In this way, the weights of a₁₂, a_1n, and a_2n must be implemented during distributed averaging control to prove the plug-and-play mechanism in the cooperative structure. Another important feature is to provide a possibility for the LCs to overlap the complementary purpose of reactive power sharing and the bus voltage regulation as a trade-off, despite the nonlinearity and uncertainty of the modeling control system, disturbances, and different load characteristics (for both hierarchical and MAS structure) [21, 24, 25, 35, 40, 71, 72, 81, 97, 100, 161, 165, 209, 216].

The distributed averaging algorithm is often the basis of the control of autonomous intelligent MGs. In [25, 100], coordinated end-to-end information (such as average voltage and active power output of MG) is obtained for a trade-off between P/Q power sharing and V − f regulation. In [161], an AVIDC method based on the virtual synchronous generator is presented equipped with average consensus control to control (Q − V). However, this is a complex algorithm for voltage regulation and system stability. In [165], a secondary V − f control is proposed, which can play as a distributed averaging PI controller for trade-off between (Q − V). In [97], the averaging control-based trade-off (Q − V) scheme can limit the capacity of the bus’s voltage in an acceptable amplitude instead of average voltage regulation. In [216], a reactive power sharing technique is presented in terms of virtual shunt and series capacitor control, which recovers the system by using a fault-tolerant secondary control. The process is implied by a dynamic average consensus algorithm to compensate for the errors caused by deviations of the V and f in the IMG. In [65, 204], an averaging control scheme is presented for algorithmic time-limited global information sharing. This has acceptable transient behavior, meticulous control against uncertainties, and the proper features of restraining the disturbance. In this case, the agents indirectly use average data exchange and also a two-level fixed-time averaging control voltage and IRPS compensation strategy in [217]. It guarantees reference tracking before the desired fixed time is presented based on the sliding mode protocol and consensus-based MAS. In [40], a correction term Z_V is derived from a consensus control approach implemented by a PI controller. This distributed averaging scheme is used to adjust the adaptive virtual inductance in the MAS structure. Another advantage is the suppression of circulating current and adjusting the output voltage of each DER. Nevertheless, the method just guarantees the configuration of paralleled microsources, and also, there is no idea to maintain the voltage close to the nominal value, despite voltage regulation. Despite the requirement of the distributed averaging algorithm for local control structures, the delay caused by the data between all DERs causes time delays.

4.3. Event-Triggered Control

In a conventional hierarchical control, a huge amount of unnecessary signals transfers from jth neighbor agents to the ith reference agent and vice versa, rapidly. It causes them to hurtle; therefore, the stochastic delay appears during the dynamic response. By event-triggered control, each agent has a degree of independence to sample events in alternating periods. Essential signals are chosen from among all unnecessary signals to deliver the agent_i (yellow arrows in Figure 10). It significantly reduces unnecessary and time-consuming sampling of SG. Therefore, time delays are significantly reduced, and so the communication network capacity will be used more effectively. The sampling period is determined according to the density of the coordination task [20, 58, 60, 71, 79, 202, 203].

The start of sampling is from that moment when the measurement values of the MG deviate from their nominal values. It starts when thresholds are set for adjustable constant intervals, time-varying, or state-dependent tolerance paths [79]. It tracks the average of event time-varying reference signals in LCs affected by a group of consensus self-adaptive algorithms. This process is done according to the boundary layer approximation methods. Furthermore, it is updated by the measured error of the controllable quantities and the operating mode of the MGs.

The states of the jth neighbor agents are transferred to the ith reference agent by observing the instantaneous time information of the next event that is functionally necessary for the operation of the system. Moreover, the precision of the process is confirmed by experiments and systematic studies [109, 200]. Many of the available event-triggered mechanisms have threshold parameters that are always fixed. Nevertheless, adaptive approaches have been reported in the literature to solve problems of delay [79, 151]. The threshold parameter is adapted according to the triggering conditions with a dynamic event-triggered communication mechanism. It can be implemented by the consensus algorithm of the centralized power controller. Thus, the MAS-based attributed coordination control is established by power-balancing consensus algorithms with state-dependent thresholds [24, 202]. However, a load change event during the IRPS compensation can threaten the system stability. In [58], an approach for V − f regulation at the secondary control is reported by the event-triggered control in MAS. The outputs of the estimators approximate the actual values when the event is just triggered. The event-triggered algorithm prevents the processing of the estimated data. It can be done to reduce the time delay and bandwidth delay of the communication link. In [20, 180], synchronization signals are introduced by a classic event-triggered control to improve reactive power sharing. Nevertheless, the success of the proposed methods depends on the use of additional communication links to trigger the synchronization compensation steps. Moreover, the load changes are not foreseen during the IRPS compensation.

The virtual impedances are achieved to ensure both fundamental positive and negative sequences by $$ signal and harmonic frequencies by $$ signal for each individual microsource, where $$ is input signal of unbalanced components from DER_k=i⟶j to its nominal value U_rate.i.j as well as suppressing harmonics of nonlinear components by $$ to H_rate.i.j. Although it is done without requiring the knowledge of feeder information, it suffers from high interference of LCs, which can lead to another type of time delay.

Although the event-triggered control improves IMGs controlled by MAS, it has a complex operation, which does not guarantee the closed-loop system stability [20, 79, 180].

8.1. Comparison of AVIDC With Other Adaptive Control Approaches

The nonimpedance-based droop methods such as the angle droop method do not guarantee reactive power sharing in different loading conditions, and active power sharing is often discussed [261]. Thus, compared to other rivals, AVIDC can recover voltage deviations under different load conditions while supporting reactive power sharing simultaneously. Moreover, it guarantees the plug-and-play mechanism because it does not include a slack bus and DER reference so that all sources are parallel. In Table 5, the features of AVIDC are compared with other adaptive methods.

8.2. The Major Technical Challenges of AVIDC

9.1. Application of AVIDC in Resistive MGs

In this situation, AVIDC is required to calculate a virtual resistance, instead of virtual inductance, according to the system disturbances to compensate for the mismatched voltage drops caused by the mismatch of the output impedances. In this way, the resistance component of the virtual impedance overcomes the inductance component. This section implements a case study to provide more information about resistive MGs.

9.2. A Case Study

In this section, a case study is investigated to show how AVIDC can work in resistive MGs. It is an IMG with nominal frequency f = 50 Hz and line-to-line voltage V_L−L = 390 V, simulated using Altair PSIM software. The system is equipped with two parallel DERs with apparent power equal to 10 kVA. Each microsource has a separate control unit, similar to other units. The resources are connected through LBC links according to Figure 12.

9.2.1. Proposed Control System

The control scheme is implemented by a process to adjust the virtual impedance value. This process consists of two operating modes, including the conventional droop/boost mode and adaptive virtual impedance tuning mode. Previously, [321] proposed a similar control approach to improve reactive power sharing in inductive IMGs. The effect of adaptive virtual impedance control in a low-voltage MG with a low X/R ratio will be studied in this paper.

Normally, all sources work in constant voltage mode. The adjustment process started with a coordinating signal, which requires a LBC link. Then, all sources switch between two operating modes sequentially using internal timers. In the conventional droop/boost mode, each source implements (P − V) and (Q − ω) characteristics to find a better working point from the viewpoint of active power sharing. The equations of the conventional droop process can be recalled from Equations (31a) and (31b). After a short time, the controller enters the adaptive virtual impedance tuning mode. In this mode, the virtual resistance value is adjusted based on the results of the previous mode.

At the end of droop/boost mode, the controller saves the average value of active power (P) through sampling by sample and hold (S/H) circuit (see Figure 13). Then, an integral controller computes the required change in virtual resistance by integrating the difference between the average power, obtained by passing the instantaneous power P_i through the low-pass filter, andP_avg values, as follows:

$$ ()

where k_avi is the gain of adaptive control coefficient for improving active power sharing and ω_c is the cutoff frequency of low-pass filter. The controller of all sources is switched between two modes multiple times. Gradually, the response of both modes leads toward ideal active power sharing. The resulting waveforms can be analyzed according to Figure 14.

The parameters of MG and controller are detailed in Table 6.

Table 6. Parameters of the proposed control system.

Microgrid parameters	Value	Controller parameters	Value
Lf1 = Lf2	360.76 × 10−6	PI controller in V–I inner loops	Kpv = 1
S1 = S2	10 kVA		Kiv = 100
Cf1 = Cf2	103.6 × 10−6		Kpc = 1
Load 1	0.0693 + j0.0002 Ω		Kic = 10,000
Load 2	0.0343 + j0.0001 Ω		ωc = 62 Hz
Resistive MG		∆f	0.0.1 Hz
ZLine1	0.1226 + j0.0245 Ω	∆V	0.02 pu
ZLine2	0.0300 + j0.0060 Ω	Kavi for conventional droop	10 × 10−6
Inductive MG
ZLine1	0.0245 + j0.1226 Ω	Kavi for reverse droop	1.8 × 10−6
ZLine2	0.0060 + j0.0300 Ω

9.2.2. Simulation Results

The simulation results can be analyzed according to Figure 15. In Figure 15(a), the active power flow tends toward equal values of load sharing between the two parallel DERs using the adaptive virtual resistance control mode. As active power sharing is a function of voltage regulation, the microsources output voltage variations during the virtual resistance adaptation process can be observed in Figure 15(b). Now, the controller can adjust the voltage in an acceptable range near the nominal value. In Figures 15(c) and 15(d), reactive power and the angular frequency of microsources are illustrated, respectively. As can be seen, the reactive power sharing remained ideal and the frequency is in an acceptable range near the nominal value. In the adaptive tuning mode, the MG frequency stabilizes toward the nominal value. In addition, compared to Figure 15, Figure 14 can be studied using AVIDC in inductive IMGs.

According to Figure 14(a), active power sharing is often maintained in the droop-controlled inductive MG. Subsequently, the MG frequency also tends to the nominal values in Figure 14(b) with the allowed tolerance. Thus, the two control modes finally converge to control the reactive power sharing as shown in Figure 14(c). Also, the IMG voltage is set at the nominal value (see Figure 14(d)). It is worth mentioning that in inductive MGs, Equation (32) must be rewritten in terms of Q instead of P.

Based on the obtained results, it can be concluded that by correctly adjusting the gain of the adaptive controller in terms of P and Q, without the need to change the droop slope, an acceptable response can be obtained from AVIDC in both resistive and inductive MGs.