Distributed secondary event-triggered voltage regulation for isolated AC microgrids: A second-order integral sliding mode approach

Abstract

Through the last decade, multi-agent systems have come to the attention of researchers in various fields. Henceforth, these systems have been investigated in various control applications. Microgrids and power systems are intriguing issues to address in agent-based systems. This research proposes a secondary control scheme based on a leader–follower consensus protocol for an islanded microgrid. The scheme aims to regulate the output voltage of distributed generators and to synchronize their frequency in a distributed manner. The control system has been embedded with the physical power system through an event-triggered communication network that determines control protocol for distributed generators. An integral sliding mode method is designed for voltage control to reach a consensus in output voltage among the distributed generators. Furthermore, a pinning-based consensus tracking problem is solved for frequency synchronization. Zeno behavior of the event-triggered mechanism has been excluded. By inducing Lyapunov-based stability analysis, this article suggests a reliable secondary control in the transient operation of isolating a microgrid. The system’s characteristics are further investigated, and results are compared with existing strategies. The proposed method maintains an acceptable voltage regulation and an accurate frequency restoration.

Keywords

Multi-agent systems consensus tracking AC microgrid distributed generators droop control event-triggered mechanism sliding mode control

Introduction

Microgrids (MGs) offer promising solutions to enhance power infrastructure reliability and address global warming. Distributed generation (DG), such as wind turbines and photovoltaic cells, increases availability in conventional power systems. However, incorporating large amounts of DGs presents challenges like system complexity and the need for advanced control systems (Bevrani et al., 2022). MGs operate in grid-connected or islanded modes, enabling economic dispatch from the main grid or technical maintenance within the MG (Ahmed et al., 2020). Hierarchical structures synchronize voltage and frequency levels of DGs and optimize power flow among Distributed energy resources (DERs) (Zhou et al., 2023). A primary control layer stabilizes voltage and frequency using droop control and current sharing with PI controllers (Olivares et al., 2014). Despite its stability, the primary control layer cannot avoid steady-state deviations, necessitating a secondary control layer for precise tracking of converter-based DERs to nominal voltage and frequency values (Dragičević et al., 2015).

A secondary layer is in charge of reasonable power flow sharing and synchronization of voltage $V$ and frequency $f$ of DGs to their rated values in a way that the allowed range for DGs to deviate from the reference is $\pm 5 %$ (Photovoltaics, Dispersed Generation and Storage, Energy, 2011). Generally, the secondary control for MGs is categorized as:

Centralized approach, where all DGs transmit their data to a Central Computing Unit (CCU) to calculate a control law for all DGs separately (Bhattar and Chaudhari, 2023). One major drawback in the centralized scheme is the high cost of communications. Also, these systems are more prone to failure and system instability with a CCU (Nawaz et al., 2023),

Decentralized approach, where each DG has its local computing unit and communicates with all other DGs to generate control signals for $V - f$ restoration (Guerrero et al., 2012). Lower communication cost makes the decentralized approach more applicable in practical systems (Liang et al., 2013). This approach maintains a much higher stability margin in MGs rather than the proposed centralized schemes in the literature, though single-point failure of a DG forces local disruption, mostly leading to system instability (Khayat et al., 2018).

Distributed approach, where each DG is considered as an agent in a multi-agent system (MAS) and shares information only with a finite set of neighbors in a network defined by a communication graph among DGs (Mesbahi and Egerstedt, 2010). This feature enables systems to lower their susceptibility to temporary local failures. Such attributes of the distributed approach have led to major advances in the recent decade, where modern power systems are in possession of flexible capabilities, for example, plug-and-play (Ning et al., 2020) and robustness to local failures of controllers (Mohiuddin and Qi, 2019).

Regarding practical systems, there are limitations imposed by bandwidth and energy constraints on communication networks in MGs. Therefore, event-triggered (ET) mechanisms can be implemented to reduce unnecessary information exchange among DGs unless a triggering condition has been violated (Shi et al., 2019). Various control methods, including leader–follower consensus protocols (Lai et al., 2020), sliding mode control (SMC) methods (e.g. robust SMC (Prasad, 2023) and terminal SMC (Shotorbani et al., 2017)), and fault-tolerant adaptive schemes (Li et al., 2020), are introduced for synchronizing voltage and frequency in a distributed system. Moreover, optimization techniques such as particle swarm optimization (PSO) (Abdelhadi et al., 2024) and model predictive control (MPC) (Nagasri and Marimuthu, 2023) are investigated for optimal power allocation and frequency regulation.

Enlightened by the discussions as mentioned earlier, a distributed secondary control is designed in this article for islanded AC MGs via an event-based communication network. A second-order integral SMC method, as proposed in the work by Ning et al. (2020), regulates the voltages of inverter-based DGs. Regardless of secondary control methods in the works by Bhattar and Chaudhari (2023) and Guerrero et al. (2012), this paper focuses on the regulation of voltage level of DGs by manipulating the derivatives of DGs’ voltages instead of regulating the voltage level itself. Thus, reactive power sharing is more rational than the conventional droop-based controller. Moreover, frequency restoration of DGs is obtained through a pinning-based control protocol. A communication graph is formed to illustrate the control system interconnections among DGs. Despite the leaderless consensus problem in the work by Dehkordi et al. (2016), in this article, reference values are broadcasted to those DGs with information access to the leader of the graph. Zeno occurrence has also been excluded. The main contributions of this paper are elaborated as follows:

This article investigates the voltage regulation problem to preserve the voltage deviation of DGs within a $5 %$ change from the virtual leader’s reference. The operability within the acceptable range for secondary voltage controllers with first-order dynamics has not been successful in the face of parameter changes in the system. Inspired by the second-order integral sliding mode in Yao et al. (2020), a voltage controller is designed to force the deviations to be desirably bounded. Voltage and its derivative for each DG are integrated in the secondary voltage controller design. Note that the work in Yao et al. (2020) focuses on single-integrator multi-agent control, while we modify the novelty of the work for determining the nominal voltage in the droop control of AC MGs.

The proposed ET mechanism for voltage regulation permits a minority set of DG(s) to update their nominal voltage in droop control at a lower rate than the others. In fact, only the DG which has access to leader’s information is required to modify its control input consistently throughout the isolated mode of the MG. The simulation results show that if the physical system faces load change, the corresponding controllers react proportionally to settle the power-sharing problem for the new load demand. Distinctly, the scheme can be modified to larger MASs with at least two neighboring agents of the leader, as long as the corresponding graph is strongly connected and balanced.

The remainder of this article is organized as follows. The second section represents with preliminaries and problem formulation. In the third section, the primary strategy of the proposed secondary control for MGs is detailed. In the fourth section, the efficacy of the designed controllers is manifested with a four-DG MG through different scenarios through simulations, that is, load change, plug-and-play capability, and communication failure. The last section concludes the article.

Preliminaries and problem formulation

In this section, the required preliminaries, graph theory fundamentals for communication topology, and the problem formulation for the secondary control of MGs are explained.

Notation

Let 1n denote the n-dimensional vector with all entries being 1, $I_{n} \in R^{n \times n}$ denote the identity matrix, and ⊗ denote the Kronecker product. For $x \in R^{n}$ , $‖ x ‖$ denotes the Euclidean norm, that is, $‖ x ‖ = ‖ x ‖_{2}$ . For $x_{i} \in R^{n_{i}}$ , $i = 1, \dots, m$ , $col (x_{1}, \dots, x_{m}) = [x_{1}^{T}, \dots, x_{m}^{T}]^{T}$ , where $x_{i}^{T}$ is the transpose of $x_{i}$ .

Directed communication

In this paper, a directed graph $G = (V, E)$ is used to describe the information exchange between DGs, where $V = {1, 2, \dots, N}$ is the agent set and $E \subset V \times V$ is the edge set. $(j, i) \in E$ represents that $D G_{i}$ can receive information from $D G_{j}$ . A sequence of edges of the form $(i, j), (j, k), \dots$ is known as a path. If there exists a path from node $i$ to node $j$ , then node $j$ is said to be reachable from node $i$ . A directed graph is strongly connected if there exists a path between each pair of agents. For graph $G$ , the adjacency matrix $A = [a_{ij}] \in R^{N \times N}$ is defined such that $a_{ii} = 0, a_{ij} > 0$ if $(j, i) \in E$ and $a_{ij} = 0$ otherwise. The Laplacian matrix $L = [l_{ij}] \in R^{N \times N}$ is defined as $l_{ii} = \sum_{j = 1}^{N} a_{ij}$ , and $l_{ij} = - a_{ij}$ for any $i \neq j$ . A balanced directed graph is where edges among two vertices have the same weight, that is, $a_{ij} = a_{ji}$ .

Furthermore, the reference values for voltage and frequency can be transmitted by a virtual leader to some of the DGs. Let $\bar{G}$ be the directed communication graph among $n$ DGs and a virtual leader. It is assumed that the leader does not receive information from other DGs in the communication graph. As $D G_{i}$ has access to leader’s information, $b_{i}$ is considered to be positive in $B = diag (b_{1}, b_{2}, \dots, b_{n})$ , and if $j$ th DG does not receive data from the virtual leader, then $b_{j} = 0$ .

State-space model for isolated AC MGs

In an MG, each DG contains a DC voltage source, a voltage source inverter (VSI), an LC filter, and an RL output connecter. The block diagram shown in Figure 1 depicts the control structure for a DG. To maintain the stability of frequency and voltage components in converter-based generators, droop control is introduced as the first control layer. For the $i$ th DG, it is noted as

\begin{matrix} ω_{i} = ω_{i}^{n} - m_{i}^{P} P_{i} \\ v_{i}^{od} = v_{i}^{n} - n_{i}^{Q} Q_{i} \end{matrix}

(1)

where $ω_{i}$ and $v_{i}^{od}$ are the rotation frequency and the voltage level of $D G_{i}$ , respectively. The main objective of the proposed secondary layer is to determine the $ω_{i}^{n}$ and $v_{i}^{n}$ , which are the droop control nominal values for $D G_{i}$ . Active power, $P_{i}$ , and reactive power, $Q_{i}$ , are the calculated power components of $D G_{i}$ . $m_{i}^{P}$ and $n_{i}^{Q}$ are the DG $i$ ’s droop coefficients. The designed voltage control method proposes the modification of $v_{i}^{n}$ such that the output voltage of $D G_{i}$ , $v_{i}^{od}$ , is synchronized to a standard $\pm 5 %$ range from the reference voltage level $v_{ref}$ . In addition, the proposed distributed frequency synchronization scheme aims to modify $ω_{i}^{n}$ such that $ω_{i}$ is restored to $ω_{ref}$ .

Based on the works by Marwali and Keyhani (2004) (primary controller model), Pogaku et al. (2007) (MG’s complete state-space model, including DGs, line currents, and loads), and Bidram et al. (2013) (feedback linearization method for the output controller of voltage and frequency), the corresponding large-signal model for $D G_{i}$ can be described in the form of

\begin{matrix} {\overset{\cdot}{x}}_{i} = f_{i} (x_{i}) + g_{i} (x_{i}) u_{i} \\ y_{i} = h_{i} (x_{i}) + d_{i} (x_{i}) u_{i} \end{matrix}

(2)

where $x_{i} = {[\begin{matrix} δ_{i} & P_{i} & Q_{i} & ξ_{dqi}^{T} & \partial_{dqi}^{T} & i_{ldqi}^{T} & v_{odqi}^{T} & i_{odqi}^{T} \end{matrix}]}^{T}$ is the state vector of the $i$ th DG, and $ξ_{dqi} = {[\begin{matrix} ξ_{di} & ξ_{qi} \end{matrix}]}^{T}$ and $\partial_{dqi} = {[\begin{matrix} \partial_{di} & \partial_{qi} \end{matrix}]}^{T}$ are voltage and current controllers’ auxiliary variables; $i_{ldqi} = {[\begin{matrix} i_{ldi} & i_{lqi} \end{matrix}]}^{T},$ $v_{odqi} = {[\begin{matrix} v_{odi} & v_{oqi} \end{matrix}]}^{T}$ , $i_{odqi} = {[\begin{matrix} i_{odi} & i_{oqi} \end{matrix}]}^{T}$ are vectors of state variables in $d - q$ frame. $f_{i} (x_{i}), g_{i} (x_{i})$ , and $h_{i} (x_{i})$ are introduced in Appendix A.

Problem formulation

Considering the state-space model of an isolated AC MG described in Appendix A, two problems are to be addressed. The first problem is to control output voltage, $y_{i} = v_{i}^{od}$ , by defining the controller in equation (2) as $u_{i} = v_{i}^{n}$ . In other words, to achieve the voltage regulation as

lim_{t \to \infty} | v_{i}^{od} (t) - v_{ref} | = 0

(3)

The second problem is to synchronize the output frequency, $y_{i} = ω_{i}$ , by setting the controller in equation (2) as $u_{i} = ω_{i}^{n}$ . The frequency restoration for $D G_{i}$ can be formulated as

lim_{t \to \infty} | ω_{i} (t) - ω_{ref} | = 0

(4)

Remark 1. The suggested distributed secondary controller in this article profoundly focuses on a novel voltage regulation scheme. Since the enforced second-order integral sliding-mode ensures asymptotic stability of the manifold (based on voltage components of DGs), the reliability of the designed voltage controller is further justified through Lyapunov stability analysis. Despite recent enhancements in the literature, few have addressed a second-order controller design for voltage regulation in MGs (Ning et al., 2020). The main focus of the conventional distributed voltage regulation protocols is to design a consensus-based control law that forces DGs to track a reference value through a leader, or leaderless averaging techniques, as in the work by Lai et al. (2020). Hence, this article introduces a scheme that includes voltage derivatives of DGs in a pinning-based controller design.

For the proposed distributed secondary control, the controller design requires each DG to exchange information with its neighbors through a balanced directed communication network. The following assumption establishes the argument of our main results in section “Distributed secondary control of MGs through event-triggered communication network.” In addition, Lemma 1 provides an essential condition for the manifestation of agent-based control in the stability analysis of the MG system.

Assumption 1. The directed graph $\bar{G}$ contains a spanning tree with the virtual leader as the root.

Lemma 1 (Mesbahi and Egerstedt (2010)). Based on Assumption 1 and by investigating the eigenvalue spectrum of $M = L + B$ , it can be induced that as long as the directed communication graph is rooted-out branching (ROB), the positive definiteness of $M$ is ensured.

Distributed secondary control of MGs through ET communication network

In the subsequent section, we introduce a distributed ET SMC approach for voltage regulation. Additionally, we employ a distributed ET controller to restore system frequency. The stability of this proposed scheme is assessed using Lyapunov-based analysis. Furthermore, we validate the practical implementation of the ET mechanism by excluding Zeno behavior.

Distributed second-order integral sliding mode voltage control with ET mechanism

In this subsection, a control law is introduced for the output voltage control problem, that is, $u_{i} = v_{i}^{n}$ and $y_{i} = v_{i}^{od}$ in equation (2) for $i = 1, 2, \dots, N$ with $N$ being the number of DGs. Note that the output $v_{i}^{od}$ is not directly influenced by the input $v_{i}^{n}$ (Bidram et al., 2014), that is, $d_{i} = 0$ . As a matter of fact, $v_{i}^{n}$ emerges in the second derivative of $v_{i}^{od}$ . Henceforth, by considering the input–output feedback linearization, a second-order integral sliding manifold is proposed to force the output voltage f DGs within an accepted range from $v_{ref}$ .

Integral sliding manifold design

For $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , the proposed sliding surface for agent $i$ is as

\begin{matrix} S_{i} (t) = \sum_{j = 1}^{N} a_{ij} ({\overset{\cdot}{v}}_{i}^{od} (t) - {\overset{\cdot}{v}}_{j}^{od} (t)) + b_{i} ({\overset{\cdot}{v}}_{i}^{od} (t) - {\overset{\cdot}{v}}_{0}^{od} (t)) \\ - \sum_{j = 1}^{N} a_{ij} ({\overset{\cdot}{v}}_{i}^{E} (0) - {\overset{\cdot}{v}}_{j}^{od} (0)) - b_{i} ({\overset{\cdot}{v}}_{i}^{od} (0) - {\overset{\cdot}{v}}_{0}^{od} (0)) \\ - \int_{0}^{t} (\sum_{j = 1}^{N} a_{ij} (υ_{il} (τ) - υ_{jl} (τ)) + b_{i} υ_{il} (τ)) d τ \end{matrix}

(5)

where $o d_{0} (t) = 0$ . The state feedback control law $u_{il} (t)$ for voltage regulation is presented in equation (8). The compact form of the designed sliding surface is rewritten as

\begin{matrix} S (t) = M \otimes I_{n} ({\overset{\cdot}{v}}^{od} (t) - 1_{N} \otimes {\overset{\cdot}{v}}_{0}^{od} (t) - {\overset{\cdot}{v}}^{od} (0) + 1_{N} \otimes {\overset{\cdot}{v}}_{0}^{od} (0)) \\ - \int_{0}^{t} M \otimes I_{n} υ_{l} (τ) d τ \end{matrix}

(6)

where the surface vector $S (t) = {[\begin{matrix} S_{1} {(t)}^{T}, S_{2} {(t)}^{T}, \dots, S_{N} {(t)}^{T} \end{matrix}]}^{T}$ and control law vector $υ_{l} (t) = {[\begin{matrix} υ_{1 l} {(t)}^{T}, υ_{2 l} {(t)}^{T}, \dots, υ_{Nl} {(t)}^{T} \end{matrix}]}^{T}$ are presented for all the DGs. If the voltage output $v_{i}^{od}$ stays on the sliding manifold, one has $S_{i} = {\overset{\cdot}{S}}_{i} = 0$ . ${\overset{\cdot}{S}}_{i} = 0$ indicates ${\overset{\cdot\cdot}{v}}^{od} - 1_{N} \otimes {\overset{\cdot\cdot}{v}}_{0}^{od} = υ_{l} (t)$ . As a result, sliding mode dynamics for $i = 1, 2, \dots, N$ is achieved as follows

{\overset{\cdot}{φ}}_{i} (t) = υ_{il} (t)

(7)

where $φ_{i} (t) = {\overset{\cdot}{v}}_{i}^{od} (t) - {\overset{\cdot}{v}}_{0}^{od} (t)$ .

ET sliding mode voltage controller design

In this subsection, the distributed integral sliding mode controller for the closed-loop dynamics of output voltages of all DGs is analyzed based on an ET mechanism. In the distributed ET scheme, let ${t_{k}^{i}}_{k = 0}^{\infty}$ be the triggering instants produced by an event-based process. The control protocol is transmitted at $t = t_{k}^{i}$ instants and holds the same value until the further triggering instant $t_{k + 1}^{i}$ using zero-order hold (ZOH). With a discontinuous enforcement of control signal, for all $t \in [t_{k}^{i}, t_{k + + 1}^{i})$ , measurement errors $ε_{1 i} (t) = v_{i}^{od} (t_{k}^{i}) - v_{i}^{od} (t), ε_{2 i} (t) = {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) - {\overset{\cdot}{v}}_{i}^{od} (t)$ are defined. Without loss of generality, it is assumed $t_{0}^{i} = 0, \forall i$ . Then, the distributed state-feedback linear voltage controller $υ_{il} (t)$ is designed as

\begin{matrix} υ_{il} (t) = C {\sum_{j \in N_{i}} a_{ij} [v_{j}^{od} (t_{k}^{j}) - v_{i}^{od} (t_{k}^{i}) + {\overset{\cdot}{v}}_{j}^{od} (t_{k}^{j}) - {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i})] \\ - b_{i} [v_{i}^{od} (t_{k}^{i}) - v_{0}^{od} (t_{k}^{i}) + {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) - {\overset{\cdot}{v}}_{0}^{od} (t_{k}^{i})]} \end{matrix}

(8)

where $N_{i}$ is a finite set of neighbors for $D G_{i}$ and $t_{k}^{j}$ denotes the last triggering instant of the $i$ th DG’s neighbors. The controller gain $C > 0$ is to be selected. The distributed ET sliding mode voltage controller is proposed as follows

\begin{matrix} u_{i}^{v} (t) = υ_{il} (t) + υ_{isw} (t), t \in [t_{k}^{i}, t_{k + 1}^{i}) \\ υ_{isw} (t) = - (ς_{1} ‖ v_{i}^{od} (t_{k}^{i}) ‖ + ς_{2} ‖ {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) ‖ + δ_{1} + δ_{2}) \\ \times sgn (S_{i} (t)) \end{matrix}

(9)

with $δ_{1}, δ_{2}, ς_{1}, ς_{2} > 0$ . Before proceeding to Theorem 1, a triggering function is proposed as

\begin{matrix} F_{i} (t) = ‖ ε_{1 i} (t) ‖^{2} + ‖ ε_{2 i} (t) ‖^{2} - μ_{i} ‖ \sum_{j = 1}^{N} a_{ij} (v_{j}^{od} (t) - v_{i}^{od} (t)) ‖^{2} \\ - μ_{i} ‖ \sum_{j = 1}^{N} a_{ij} ({\overset{\cdot}{v}}_{j}^{od} (t) - {\overset{\cdot}{v}}_{i}^{od} (t)) ‖^{2} - ξ_{i} e^{- ϱ_{i} t} \\ - μ_{i} b_{i} [{‖ v_{0}^{od} - v_{i}^{od} (t) ‖}^{2} + {‖ {\overset{\cdot}{v}}_{i}^{od} (t) ‖}^{2}] \end{matrix}

(10)

where $μ_{i}, ξ_{i}, ϱ_{i} > 0$ . As long as $F_{i} (t) \geq 0$ , no triggering occurs. The following theorem states the reachability of the sliding surface for the system trajectories in equation (7).

Theorem 1. Under Assumption 1 and the distributed ET sliding mode voltage regulation protocol in equation (9) for each $D G_{i}$ , the state trajectory of second-order dynamics of output voltages in the MG will reach the sliding manifold. As long as the states stay on the surface, sliding mode states will reach asymptotic stability.

Proof. First, we consider a candidate Lyapunov function as

V_{1} (t) = \frac{1}{2} S {(t)}^{T} {(M \otimes I_{n})}^{- 1} S (t)

(11)

Taking the Lyapunov function’s derivative concerning $t$ , one has

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = S {(t)}^{T} {(M \otimes I_{n})}^{- 1} \overset{\cdot}{S} (t) \\ = S {(t)}^{T} {(M \otimes I_{n})}^{- 1} [M \otimes I_{n} ({\overset{\cdot}{v}}^{od} - 1_{N} \otimes {\overset{\cdot}{v}}_{0}^{od} - υ_{l} (t))] \\ = S {(t)}^{T} (υ_{sw} (t)) \end{matrix}

(12)

Note that the leader has a constant value and does not possess a dynamic equation. Thus, the stability of the proposed Lyapunov function is obtained as

{\overset{\cdot}{V}}_{1} (t) \leq - \sum_{i = 1}^{N} ‖ S_{i} (t) ‖ (ς_{1} ‖ v_{i}^{od} (t_{k}^{i}) ‖ + ς_{2} ‖ {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) ‖ + δ_{1} + δ_{2})

(13)

Accordingly, for $t \in [t_{k}^{i}, t_{k + 1}^{i})$ at times when $S (t) \neq 0$ , ${\overset{\cdot}{V}}_{1} (t) < 0$ , with a proper selection of $δ_{1}, δ_{2}, ς_{1}, ς_{2} > 0$ . So, the output voltage of all DGs and their derivatives will reach the sliding manifold within a finite time. That will conclude the proof. □

Remark 2. Based on the discussion above, when the states of all DGs arrive at the designed sliding manifold, $S_{i} (t) = {\overset{\cdot}{S}}_{i} (t) = 0$ implies that the proposed method effectively compensates external disturbances with the switching term in control input $υ_{isw} (t)$ as described in equation (9).

Remark 3. The ET second-order integral sliding mode voltage controller proposed in equation (9) is inherently non-smooth due to the inclusion of the sign function in the SMC approach. However, to mitigate chattering behavior during practical implementation of the control law, an approximation of the sign function is represented as $sign S (t) \approx [S (t) / (‖ S (t) ‖) + ϵ]$ and is commonly used, where $ϵ$ represents a small positive constant.

Remark 4. To establish a detailed comprehension of the sliding manifold in equation (5), we ought to discuss the design in the integral sliding surface. The first term stands for the disagreement that needs to reach zero. The second term ensures that the initial discrepancies in voltage derivatives are accounted for, providing a baseline for the control protocol. The integral term helps in mitigating steady-state error and improving the robustness of the control system against disturbances and uncertainties.

As the output voltage of $D G_{i}$ and its derivative enter the integral sliding surface, one has $S_{i} = {\overset{\cdot}{S}}_{i} = 0$ . With ${\overset{\cdot}{S}}_{i} = 0$ , disagreement dynamics of the sliding mode controller for voltage regulation is achieved. The second-order dynamics of voltage and its derivative for the $i$ th DG can be rewritten as

\begin{matrix} {\overset{\cdot}{γ}}_{i} (t) = φ_{i} (t) \\ {\overset{\cdot}{φ}}_{i} (t) = C \sum_{j \in N_{i}} a_{ij} [v_{j}^{od} (t_{k}^{j}) - v_{i}^{od} (t_{k}^{i}) + {\overset{\cdot}{v}}_{j}^{od} (t_{k}^{j}) - {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i})] \\ - C b_{i} [v_{i}^{od} (t_{k}^{i}) - v_{0}^{od} + {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i})] \end{matrix}

(14)

where $γ_{i} (t) = v_{i}^{od} (t) - v_{0}^{od}$ and $φ_{i} (t) = {\overset{\cdot}{v}}_{i}^{od} (t) - {\overset{\cdot}{v}}_{0}^{od} = {\overset{\cdot}{v}}_{i}^{od} (t)$ . The dynamics in equation (14) can be reformulated to manifest the error dynamics as

\begin{matrix} {\overset{\cdot}{γ}}_{i} (t) = φ_{i} (t) \\ {\overset{\cdot}{φ}}_{i} (t) = C \sum_{j \in N_{i}} a_{ij} [e_{1 j} (t) + γ_{j} (t) - (ε_{1 i} (t) + γ_{i} (t)) + ε_{2 j} (t) \\ + φ_{j} (t) - (ε_{2 i} (t) + φ_{i} (t))] - C b_{i} (ε_{1 i} (t) \\ + γ_{i} (t) + ε_{2 i} (t) + φ_{i} (t)) \end{matrix}

(15)

which the compact form of equation (15) is further deduced as

\begin{matrix} \overset{\cdot}{γ} (t) = φ (t) \\ \overset{\cdot}{φ} (t) = - C M \otimes I_{n} (ε_{1} (t) + ε_{2} (t) + γ (t) + φ (t)) \end{matrix}

(16)

Event errors are defined as $ε_{1 i} (t) = v_{i}^{od} (t_{k}^{i}) - v_{i}^{od} (t), ε_{2 i} (t) = {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) - {\overset{\cdot}{v}}_{i}^{od} (t), ε_{1 j} (t) = v_{j}^{od} (t_{k}^{j}) - v_{j}^{od} (t)$ and $ε_{2 j} (t) = {\overset{\cdot}{v}}_{j}^{od} (t_{k}^{j}) - {\overset{\cdot}{v}}_{j}^{od} (t)$ for $i, j = 1, 2, \dots, N$ . In the next section, stability analysis for the proposed distributed voltage regulation scheme is to be addressed.

Stability evaluation

Considering the distributed ET sliding mode voltage controller in equation (9), the sufficient condition of the leader–follower consensus is obtained for the sliding mode dynamics of the output voltage of all DGs. Hence, the following theorem is introduced to ensure the asymptotic stability of the system in equation (15).

Theorem 2. Under the distributed ET control protocol in equation (9) and the trigger function in equation (10), the consensus-based voltage regulation for the obtained sliding mode dynamics in equation (16) is achieved if the following criteria hold:

\begin{matrix} 0 < λ_{\min} (Ω) - 2 \sqrt{α μ ζ} ∥ C M ∥ - ϵ ∥ C M ∥ \\ 1 < 2 C λ_{\min} (M), \\ Ω = [\begin{matrix} C M & 0 \\ 0 & C M - I_{N} \end{matrix}] \otimes I_{n} \end{matrix}

where $ζ = λ_{\max} (LL + B), α > 0, ϵ > 0$ and $μ = mi n_{i} {μ_{i}}$ .

Proof. Regarding Lemma 1 and the results from Theorem 1, we introduce the following Lyapunov function

V_{2} (t) = V_{1} (t) + \frac{1}{2} y^{T} (t) Θ y (t)

(17)

where $y (t) = {[\begin{matrix} γ^{T} (t) & φ^{T} (t) \end{matrix}]}^{T}, Θ = [\begin{matrix} 2 C M & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n}$ .

As $S (t) = \overset{\cdot}{S} (t) = 0$ , the positive definiteness of the candidate Lyapunov functional is guaranteed regarding $λ_{\min} (2 C M - I_{N}) > 0$ for $y (t) \neq 0$ . Considering the sliding mode dynamics in equation (16), the differentiation of $V_{2} (t)$ is achieved as

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) = {\overset{\cdot}{V}}_{1} (t) + y^{T} (t) [\begin{matrix} 2 C M & I_{N} \\ I_{N} & I_{N} \end{matrix}] \otimes I_{n} \overset{\cdot}{y} (t) \\ = {\overset{\cdot}{V}}_{1} (t) - γ^{T} (t) C M \otimes I_{n} γ (t) \\ - φ^{T} (t) C M \otimes I_{n} φ (t) + φ^{T} (t) φ (t) \\ - C (γ^{T} (t) + φ^{T} (t)) M \otimes I_{n} (ε_{1} (t) + ε_{2} (t)) \\ = {\overset{\cdot}{V}}_{1} (t) - y^{T} (t) [\begin{matrix} C M & 0 \\ 0 & C M - I_{N} \end{matrix}] \otimes I_{n} y (t) \\ - y^{T} (t) [\begin{matrix} C M & C M \\ C M & C M \end{matrix}] \otimes I_{n} ε (t) \end{matrix}

(18)

where $ε (t) = {[\begin{matrix} ε_{1}^{T} (t) & ε_{2}^{T} (t) \end{matrix}]}^{T}$ .

According to the event function in equation (10), one has

\begin{matrix} ∥ ε (t) ∥^{2} = \sum_{i = 1}^{N} (∥ ε_{1 i} (t) ∥^{2} + ∥ ε_{2 i} (t) ∥^{2}) \\ \leq α μ [γ^{T} (t) (LL \otimes I_{n}) γ (t) + φ^{T} (t) (LL \otimes I_{n}) φ (t)] \\ + α μ [γ^{T} (t) (B \otimes I_{n}) γ (t) + φ^{T} (t) (B \otimes I_{n}) φ (t)] \\ + \sum_{i = 1}^{N} α ξ_{i} e^{- ϱ_{i} t} \\ = α μ y^{T} (t) [I_{2} \otimes (LL \otimes I_{n})] y (t) \\ + α μ y^{T} (t) [I_{2} \otimes (B \otimes I_{n})] y (t) + \sum_{i = 1}^{N} α ξ_{i} e^{- ϱ_{i} t} \\ \leq α μ ζ ∥ y (t) ∥^{2} + \sum_{i = 1}^{N} α ξ_{i} e^{- ϱ_{i} t} \end{matrix}

(19)

Based on Young’s inequality (Sababheh and Choi, 2016) $(ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q})$ for $p, q < 1$ and any $ϵ > 0$ , the derivative of $V_{2} (t)$ can be simplified as

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) = - y^{T} (t) [\begin{matrix} C M & 0 \\ 0 & C M - I_{N} \end{matrix}] \otimes I_{n} y (t) \\ - y^{T} (t) [\begin{matrix} C M & C M \\ C M & C M \end{matrix}] \otimes I_{n} ε (t) \\ \leq - y^{T} (t) [\begin{matrix} C M & 0 \\ 0 & C M - I_{N} \end{matrix}] \otimes I_{n} y (t) \\ + 2 ∥ y (t) ∥ ∥ M ∥ ∥ ε (t) ∥ \\ \leq - λ_{\min} (Ω) ∥ y (t) ∥^{2} + \frac{1}{ϵ} α ∥ C M ∥ \sum_{i = 1}^{N} ξ_{i} e^{- ϱ_{1} t} \\ + 2 \sqrt{α μ ζ} ∥ C M ∥ ∥ y (t) ∥^{2} + ϵ ∥ C M ∥ ∥ y (t) ∥^{2} \\ \leq - σ V_{1} (t) + \frac{1}{ϵ} (α ∥ C M ∥) \sum_{i = 1}^{N} ξ_{i} e^{- ϱ_{l} t} \\ σ = \frac{2 (λ_{\min} (Ω) - 2 \sqrt{α μ ζ} ∥ C M ∥ - 2 ϵ ∥ C M ∥)}{λ_{\max} (Θ)} > 0 \end{matrix}

(20)

According to Barbalat’s lemma (Singh and Khalil, 2005), $lim_{t \to \infty} ‖ y ‖ = 0$ is achieved. Thus, it is inferred that the leader–follower consensus for the sliding mode dynamics for the output voltage of DGs in an MG is achieved. Thus, the voltage derivatives arrive at the designed sliding surface in equation (6). Henceforth, the voltage regulation of all DGs is attained. This completes the proof. □

Zeno behavior exclusion

Zeno behavior is known as the occurrence of an infinite number of triggering instants in a finite period. In this section, adequate reasoning is provided to exclude such an undesirable phenomenon. To avert Zeno behavior, one must acknowledge that in the triggering sequence ${t_{k}^{i}}_{k = 0}^{\infty}$ , the intervention period $τ_{k}^{i} = t_{k + 1}^{i} - t_{k}^{i}$ has to be positive. The following theorem investigates this subject.

Theorem 3. In the distributed ET sliding mode voltage regulator in equation (9), considering the event generator in equation (10) for $D G_{i}$ , $i = 1, 2, \dots, N$ , the intervention time $τ_{k}^{i}$ between two separated triggering moments is always a positive value.

Proof. According to the event function in equation (10), for all $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , one has

\begin{matrix} ∥ ε_{1 i} (t) ∥^{2} + ∥ ε_{2 i} (t) ∥^{2} \leq μ_{i} ‖ \sum_{j = 1}^{N} a_{ij} (v_{j}^{od} (t) - v_{i}^{od} (t)) ‖^{2} \\ + μ_{i} ‖ \sum_{j = 1}^{N} a_{ij} ({\overset{\cdot}{v}}_{j}^{od} (t) - {\overset{\cdot}{v}}_{i}^{od} (t)) ‖_{i}^{2} + ξ_{i} e^{- ϱ_{i} t} \\ + μ_{i} b_{i} [{‖ v_{0}^{od} - v_{i}^{od} (t) ∥^{2} + ∥ {\overset{\cdot}{v}}_{i}^{od} (t) ‖}^{2}] \end{matrix}

(21)

The upper right-hand derivative of the terms $‖ ε_{1 i} (t) ‖ + ‖ ε_{2 i} (t) ‖$ for each $D G_{i}$ is obtained as follows

\begin{matrix} D^{+} (∥ ε_{1 i} (t) ∥ + ∥ ε_{2 i} (t) ∥) \\ \leq (1 + ς_{1} + ς_{2}) (∥ ε_{1 i} (t) ∥ + ∥ ε_{2 i} (t) ∥) + ξ (t_{k}^{i}, t_{k}^{j}) \\ Φ (t_{k}^{i}, t_{k}^{j}) = (1 + \sqrt{n}) ς_{1} ‖ v_{i}^{od} (t_{k}^{i}) ‖ + (1 + \sqrt{n}) ς_{2} ‖ {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) ‖ \\ + ‖ {\overset{\cdot}{v}}_{i}^{od} (t_{k}^{i}) ‖ + \sqrt{n} (δ_{1} + δ_{2}) + ∥ υ_{il} ∥ \end{matrix}

(22)

From equation (21), it can be induced that

\begin{matrix} ‖ ε_{1 i} (t) ‖ + ‖ ε_{2 i} (t) ‖ \leq \frac{Φ (t_{k}^{i}, t_{k}^{j})}{1 + ς_{1} + ς_{2}} (e^{(1 + ς_{1} + ς_{2}) (t - t_{k}^{i})} - 1), \end{matrix}

(23)

then we extend inequality (23) to the event-triggering condition in equation (10), which leads to

\begin{matrix} ∥ ε_{1 i} (t_{k + 1}^{i}) ∥^{2} + ∥ ε_{2 i} (t_{k + 1}^{i}) ∥^{2} = Ψ (t_{k + 1}^{i}) + ξ_{i} e^{- ϱ_{i} t_{k + 1}^{i}} \\ Ψ (t_{k + 1}^{i}) = μ_{i} ∥ \sum_{j = 1}^{N} a_{ij} (v_{j}^{od} (t_{k + 1}^{i}) - v_{i}^{od} (t_{k + 1}^{i})) ∥^{2} \\ + μ_{i} b_{i} ∥ v_{0}^{od} - v_{i}^{od} (t_{k + 1}^{i}) ∥^{2} \\ + μ_{i} ∥ \sum_{j = 1}^{N} a_{ij} ({\overset{\cdot}{v}}_{j}^{od} (t_{k + 1}^{i}) - {\overset{\cdot}{v}}_{i}^{od} (t_{k + 1}^{i})) ∥^{2} \\ + μ_{i} b_{i} ∥ {\overset{\cdot}{v}}_{i}^{od} (t_{k + 1}^{i}) ∥^{2} \end{matrix}

(24)

Consequently, the following inequality is obtained

\begin{matrix} 0 < ξ_{i} e^{- ϱ_{i} t_{k + 1}} \leq Ψ (t_{k + 1}^{i}) + ξ_{i} e^{- ϱ_{i} t_{k + 1}^{i}} \\ \leq (∥ ε_{1 i} (t_{k + 1}^{i}) ∥ + ∥ ε_{2 i} (t_{k + 1}^{i}) ∥ +) \\ \leq {[\frac{Φ (t_{k}^{i}, t_{k}^{j})}{1 + ς_{1} + ς_{2}} (e^{(1 + ς_{1} + ς_{2}) (t_{k + 1}^{i} - t_{k}^{i})} - 1)]}^{2} . \end{matrix}

(25)

With a proper selection of $ξ_{i} > 0$ , the positivity of $τ_{k}^{i}$ for all $D G_{i}$ is assured and Zeno occurrence is avoided. The proof is now completed. □

To provide a nominal voltage value in equation (1) that can share the demanded reactive power of the local loads among DGs in the MG, we propose a distributed reactive power controller in the form of $u_{i}^{Q} = Q_{i, \max} \sum_{j \in N_{i}} a_{ij} (t) [\frac{Q_{j} (t)}{Q_{j, \max}} - \frac{Q_{i} (t)}{Q_{i, \max}}]$ , and formulating the designed nominal voltage as $v_{i}^{n} (t) = \int^{[} u_{i}^{v} (τ) + n_{i}^{Q} u_{i}^{Q} (τ)] d τ$ . The secondary voltage controller is finally designed. We now proceed to introduce the frequency controller design.

Distributed frequency restoration with ET mechanism

In the following section, we address the frequency restoration problem in secondary control of MGs, that is, $u_{i} = ω_{i}^{n}$ and $y_{i} = ω_{i}$ in equation (2). Be aware that the output frequency $ω_{i}$ is directly influenced by the input $ω_{i}^{n}$ (Bidram et al., 2013), which leads to the appearance of the input in the output in the state-space modeling of the $i$ th DG in equation (2).

For $t \in [t_{k}^{i}, t_{k + 1}^{i})$ , using the input–output feedback linearization, the distributed ET frequency controller is designed as follows

\begin{matrix} u_{i}^{ω} (t) = & K \sum_{j \in N_{i}} [a_{ij} (ω_{j} (t_{k}^{j}) - ω_{i} (t_{k}^{i})) - b_{i} (ω_{i} (t_{k}^{i}) - ω_{0})] \end{matrix}

(26)

Subsequently, we define disagreement auxiliary variable as $e_{i} (t) = ω_{i} (t) - ω_{0}$ and the vector form of the disagreement variable $e (t) = {[\begin{matrix} e_{1} (t) & e_{2} (t) & \dots & e_{n} (t) \end{matrix}]}^{T}$ , and also triggering error margin as $δ_{i}^{ω} (t) = ω_{i} (t_{k}^{i}) - ω_{i} (t)$ and its vector form $δ^{ω} (t) = {[\begin{matrix} δ_{1}^{ω} (t) & δ_{2}^{ω} (t) & \dots & δ_{n}^{ω} (t) \end{matrix}]}^{T}$ . In the following, error dynamics of the frequency controller in equation (26) are obtained as

ė (t) = - ℳ [e (t) + δ (t)] .

(27)

To investigate the stability of the proposed controller, we introduce a Lyapunov functional $V_{3} (t) = \frac{1}{2} ε^{T} ε$ , regarding $ε (t) = M e (t)$ . By differentiating $V_{2} (t)$ , one has

{\dot{V}}_{3} (t) = ε^{T} ℳ [ε (t) + ℳ δ (t)]

(28)

Before we proceed further, let us introduce the frequency triggering condition as

| δ_{i}^{ω} (t) | \leq σ^{ω} | \sum_{j \in N_{i}} a_{ij} [ω_{j} (t) - ω_{i} (t)] + b_{i} [ω_{0} - ω_{i} (t)] |

(29)

where $0 < σ^{ω} \leq κ \frac{λ_{\min} (M)}{| | M {| |}^{2}}$ and $0 < κ < 1$ . Therefore, we can investigate the differentiation of $V_{3} (t)$ as

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = ε^{T} (t) M ε (t) - ε^{T} (t) M^{2} δ (t) \\ \leq - λ_{\min} (M) ‖ ε (t) ‖^{2} + ‖ ε (t) ‖ . ‖ M ‖^{2} . ‖ σ (t) ‖ \\ - [(1 - κ) λ_{\min} (M)] ‖ ε ‖^{2} \\ = - [(1 - κ) λ_{\min} (M)] V_{3} (t) \end{matrix}

(30)

In the discussion mentioned above, equation (30) implies that the proposed distributed ET frequency controller in equation (26) solves the frequency restoration problem in equation (4).

A frequency nominal value is obtained to feed a control signal into all DGs for their frequency restoration to a rated constant and optimal distribution of active power among them. First, let us introduce an active power controller as $u_{i}^{P} = P_{i, \max} \sum_{j \in N_{i}} a_{ij} (t) [\frac{P_{j} (t)}{P_{j, \max}} - \frac{P_{i} (t)}{P_{i, \max}}]$ . The terms $u_{i}^{P}$ and $u_{i}^{ω}$ form the nominal frequency as $ω_{i}^{n} (t) = \int^{[} u_{i}^{ω} (τ) + m_{i}^{P} u_{i}^{P} (τ)] d τ$ .

Discussion

In this section, a secondary control scheme for AC MGs has been proposed. The control system allows communication among DGs through an ET communication network through interfaced converters of all DGs. To regulate voltage level of DGs, a second-order sliding mode controller was designed, compensating external disturbances with a switching capability of the DSMC’s correction term in equation (9). Then, a nominal voltage value is defined for each DG. Moreover, a distributed frequency controller was suggested in equation (26) to construct the nominal frequency for each DG. We now advance to the next section to validate the results of the proposed method.

Simulation results

In this section, two case studies are conducted on a test system with four DGs (i.e. $N = 4$ ) and another with eight DGs (i.e. $N = 8$ ). The communication graphs of both sets of DGs are depicted in Figure 2(a) and (b), respectively. The Laplacian matrices of both scenarios can be induced based on their topologies according to Figure 2. The information of the leader’s reference frequency ( $ω^{ref} = 2 π \times 50$ ) and voltage ( $v_{od}^{ref} = 311$ ) is available to DG1. The MG’s DGs’ characteristics, types of load, and power line attributes are stated in Table 1. The simulations are conducted in the Matlab/Simulink environment.

Table 1.

Parameters for DGs, loads, and lines (Lai et al., 2016).

DGs 1, 2, 7, 8		DGs 3, 4, 5, 6
$V_{DC}$	$800 V$	$V_{DC}$	$680 V$
$m^{P}$	$1.06 \times 10^{- 4}$	$m^{P}$	$0.708 \times 10^{- 4}$
$n^{Q}$	$5.02 \times 10^{- 3}$	$n^{Q}$	$5.26 \times 10^{- 3}$
$R_{c}$	$0.05 Ω$	$R_{c}$	$0.06 Ω$
$L_{c}$	$0.21 mH$	$L_{c}$	$0.175 mH$
$R_{f}$	$0.05 Ω$	$R_{f}$	$0.05 Ω$
$L_{f}$	$0.15 mH$	$L_{f}$	$0.15 mH$
$C_{f}$	$50 μ F$	$C_{f}$	$50 μ F$
$P_{\max}$	$20 kW$	$P_{\max}$	$30 kW$
$Q_{\max}$	$13.2 kVAr$	$Q_{\max}$	$22 kVAr$
$Loa d_{1, 5}$	$Loa d_{2, 6}$	$Loa d_{3, 7}$	$Loa d_{4, 8}$
$19.4 Kw$	$27.8 Kw$	$22.4 Kw$	$21.3 Kw$
$Lin e_{1, 5}$	$Lin e_{2, 6}$	$Lin e_{3, 7}$	$Lin e_{4, 8}$
$0.18 + j 2.9 Ω$	$0.34 + j 5.6 Ω$	$0.2 + j 3.2 Ω$	$0.3 + j 4.9 Ω$

The ET mechanism constants for the voltage regulation with the second-order integral sliding mode approach are $α = 0.02, ϵ = 0.0011, μ_{i} = 0.00021, ξ_{i} = 2.1, ϱ_{i} = 5.2$ with $σ > 0$ , which does not violate inequality (21). The sliding mode coefficients are $C = 5, ς_{1} = 0.1, ς_{2} = 0.2, δ_{1} = 2.8, δ_{2} = 0.001$ , in a way that equation (13) holds. These values are determined through a trial-and-error method for well-performing secondary voltage controller subsystems of all DGs. Also, the controller gain for the frequency restoration is $K = 2$ with the event-triggering constant $σ^{ω} = 0.0085$ . The initial values of all eight DGs for frequency and voltage are $ω_{1} (0) = ω_{2} (0) = ω_{3} (0) = ω_{4} (0) = ω_{5} (0) = ω_{6} (0) = ω_{7} (0) = ω_{8} (0) = 2 π \times 50$ and $v_{od 1} (0) = 311.3, v_{od 2} (0) = 311.2, v_{od 3} (0)$ $= 311, v_{od 4} (0) = 311.12, v_{od 5} (0) = 311.8, v_{od 6} (0) = 310.8,$ $v_{od 7} (0) = 310.5, v_{od 8} (0) = 312.1$ . We now evaluate the test system’s behavior under different circumstances. The simulations are conducted with the time step $T_{s} = 0.01 ms$ .

Performance evaluation: No load change

In this section, the test system’s performance is investigated without any load change. We consider two scenarios, First, a four-DG MG, and second, an MG with eight DGs to evaluate the control system’s performance at a larger scale. Figure 3(a) and (c) shows output voltage and frequency of four-DG MG, and Figure 3(b) and (d) shows output voltage and frequency of eight-DG MG. Voltage regulation and frequency synchronization are achieved at $t = 0.5$ and $t = 0.75$ , respectively. As can be seen, the acceptable $5 %$ deviation range is satisfied for both scenarios. Note that Figure 4 depicts the state trajectories of integral sliding manifolds for $N = 4$ .

Load change

In this subsection, we will investigate how the proposed method manages to cope with load changes in the grid. For $N = 4$ as in Figure 2(a), the local load of DG2 is disconnected throughout the simulation except in the time interval of $t \in [3, 5)$ . In addition, the local load of DG3 disconnects at $t = 8 s$ and re-connects at $t = 10 s$ . Voltage and frequency of all DGs are depicted in Figure 5(a) and (b), respectively. The designed method for voltage regulation modifies DGs’ voltage levels within a second to confront the alterations. The designed sliding surface’s state trajectories are depicted in Figure 6. The sign function in equation (9) compensates the load variations whenever one occurs. The load change’s effect on power sharing among DGs is perceptible, as shown in Figure 7(a) for the reactive power of DGs and Figure 7(b) for their active power sharing. The event instants for all DGs, in comparison with the continuous mode of a communication network, can be seen in Figure 8(a) (voltage) and (b) (frequency). By manifesting the triggering function regarding distributed approach in equation (10) for the voltage regulation, it can be deduced that only a DG is sufficient enough for its controllers to update their triggered values rapidly throughout the simulation process. Considering load changes, as the local load of DG2 connects at $t = 3$ and disconnects at $t = 5$ , its voltage controller transmits control signal to compensate the changes. For the frequency restoration triggering mechanism, all DGs exchange information relatively to their neighbors. In this scenario, DG2 and DG3 are triggered at higher rates rather than DG1 and DG4 in frequency restoration.

Evaluation of the proposed ET secondary control

In this section, the performance of the proposed method is evaluated with the developed ET secondary control in the work by Wang et al. (2019). Figure 9 depicts (a) the output voltage of DGs with the secondary control and (b) the frequency restoration proposed in the work by Wang et al. (2019). As can be inferred from the voltage response in Figure 9(a), the maximum difference between two separate DGs’ output voltages is over $10 p . u .$ However, the proposed secondary control for voltage regulation in this article keeps the divergence from the reference value below $1 p . u .$ , as in Figure 5(a). The frequency synchronizations in this article and the one in the work by Wang et al. (2019) provide reasonable restoration to their reference values.

The event conditions are first evaluated, and then compared to existing proposed triggering functions in the literature. The event condition (10) for voltage regulation is designed based on second-order tracking problem, thus DGs are required to update their control input at lower rates. On the other hand, the frequency controllers are excited by the triggering function (29) with first-order tracking dynamics. Influenced by the order reduction in defining the event condition, and the fact that DGs synchronize their frequency output simultaneously, there are more triggering instants in comparison with the voltage controllers.

To evaluate the event-triggering conditions, Figure 10 depicts triggering instants of the proposed ET mechanism with the investigated method in the work by Wang et al. (2019). In this scenario, the simulation is executed through $[0, 60)$ interval. The communication graphs are as in Figure 2(a). The event-related parameters for voltage controllers are selected as $μ_{1, 3} = 0.00021, μ_{2, 4} = 0.0051, ξ_{1, 3} = 2.1, ξ_{1, 3} = 1.9, ϱ_{1, 3} = 5.2$ , and $ϱ_{2, 4} = 3.5$ . The event variables for frequency controllers are chosen as $σ_{1, 3}^{ω} = 0.0085$ and $σ^{ω} = 0.0105$ . The local load of DG3 disconnects at $t = 20 s$ , and the local load of DG2 connects after $t = 40 s$ . By investigating the results, it can be induced that the designed voltage controller (9) alongside the distributed event function (10) leads a significant reduction in event generation compared to Figure 10(b). In the proposed voltage event condition, only the DG1 with access to the virtual leader needs to update rapidly. Moreover, when DG3’s local load disconnects, it contributes more to the reactive power sharing to shred the other DGs’ reactive supply rate. Comparing frequency-triggering instants, both ET mechanisms generate approximately similar results, with a small distinction. The proposed frequency controller spreads the triggering instants through simulation time, while the event instants in Figure 10(d) induce that triggers occur at large intervals and DGs’ events start almost similarly.

Conclusion

In this article, a distributed secondary control has been investigated for isolated AC MGs regarding direct communication networks among DGs. To reduce communication costs, an event-triggering mechanism is enforced to omit unnecessary information exchange. To regulate the output voltage of converter-based DGs, an ET second-order integral sliding mode controller is proposed to compensate for the droop control’s deficiency in steady-state performance. Furthermore, mean-square synchronization of frequency is developed to construct the frequency controller in the suggested distributed secondary scheme. Note that Zeno behavior of the event-triggering mechanism has been excluded. The evaluation of how the proposed secondary control performs is done through different scenarios to probe the system’s capability of handling load change and communication loss. For different scenarios, it is inferred that the designed voltage controller maintains the voltage level of all DGs within a $0.2 %$ range of the voltage reference value. Furthermore, frequency synchronization has driven DGs to vary $0.5 %$ from their rated value at the most.

Footnotes

Appendix A

In this section, the deduction of $f_{i} (x_{i}), g_{i} (x_{i}), h_{i} (x_{i})$ , and $d_{i} (x_{i})$ in equation (2) is elaborated. Due to the fact that the state-space matrices are too large, the inference of system matrices is left to readers, and also can be found in the work by Pogaku et al. (2007). The elaboration of the MG’s modeling is inspired by the work by Ning et al. (2020).

The difference between the angle of the reference frame for $D G_{i}$ and the common reference frame is denoted as $δ_{i}$ . The dynamics of $δ_{i}$ is defined as

(31)

\begin{matrix} \overset{\cdot}{δ} = ω_{i} - ω_{com} \end{matrix}

where $ω_{i}$ is the angular frequency of the $D G_{i}$ ’s reference frame and $ω_{com}$ is the angular frequency of the chosen common reference frame among all DGs.

The power controller dynamics are stated as

(32)

\begin{matrix} {\overset{\cdot}{P}}_{i} = - ω_{ci} P_{i} + ω_{ci} (v_{i}^{od} i_{i}^{od} + v_{i}^{oq} i_{i}^{oq}) \\ {\overset{\cdot}{Q}}_{i} = - ω_{ci} Q_{i} + ω_{ci} (v_{i}^{oq} i_{i}^{od} - v_{i}^{od} i_{i}^{oq}) \end{matrix}

where $v_{i}^{od}, i_{i}^{od}, v_{i}^{oq}, i_{i}^{oq}$ are the $d - q$ components of the output voltage $v_{i}^{o}$ and output current $i_{i}^{o}$ , $ω_{ci}$ is the cutoff frequency of the low-pass filter; $P_{i}$ and $Q_{i}$ denote the output active and reactive powers, respectively.

According to the fact that the output voltage of each DG is aligned with the $d$ -axis of the direct and quadratic frame (Vo-Duy and Ta, 2023), one obtains

(33)

\begin{matrix} v_{i}^{* od} = v_{i}^{n} - n_{i}^{Q} Q_{i} \\ v_{i}^{* oq} = 0 \end{matrix}

where $v_{i}^{* od}$ and $v_{i}^{* oq}$ are $d - q$ voltage setpoints of $D G_{i}$ and $n_{i}^{Q}$ is its droop coefficient.

The voltage and current controller dynamics of the primary control level are

(34)

\begin{matrix} {\overset{\cdot}{ξ}}_{d} = v_{od}^{*} - v_{od} \\ {\overset{\cdot}{ξ}}_{q} = v_{oq}^{*} - v_{oq} \\ i_{ld}^{*} = F i_{od} - ω_{n} C_{f} v_{oq} + K_{pv} (v_{od}^{*} - v_{od}) + K_{iv} ξ_{d} \\ i_{lq}^{*} = F i_{oq} - ω_{n} C_{f} v_{od} + K_{pv} (v_{oq}^{*} - v_{oq}) + K_{iv} ξ_{q} \\ {\overset{\cdot}{\partial}}_{d} = i_{ld}^{*} - i_{ld} \\ {\overset{\cdot}{\partial}}_{q} = i_{lq}^{*} - i_{lq} \\ v_{id}^{*} = - ω_{n} L_{f} i_{lq} + K_{pc} (i_{ld}^{*} - i_{ld}) + K_{ic} \partial_{d} \\ v_{iq}^{*} = ω_{n} L_{f} i_{ld} + K_{pc} (i_{lq}^{*} - i_{lq}) + K_{ic} \partial_{q} \end{matrix}

where $ξ_{d}, ξ_{q}, \partial_{d}, \partial_{q}$ are the auxiliary primary-level control variables for voltage and current in the $d - q$ frame, and $ω_{n}$ is nominal angular frequency.

The output LC filter and coupling inductor dynamics are

(35)

\begin{matrix} \frac{d i_{ld}}{dt} = \frac{- r_{c}}{L_{f}} i_{ld} + ω_{0} i_{lq} + \frac{1}{L_{f}} v_{id} - \frac{1}{L_{f}} v_{od} \\ \frac{d i_{lq}}{dt} = \frac{- r_{c}}{L_{f}} i_{lq} - ω_{0} i_{ld} + \frac{1}{L_{f}} v_{iq} - \frac{1}{L_{f}} v_{oq} \\ \frac{d v_{od}}{dt} = ω_{0} i_{oq} + \frac{1}{C_{f}} i_{ld} - \frac{1}{C_{f}} i_{od} \\ \frac{d v_{oq}}{dt} = - ω_{0} i_{od} + \frac{1}{C_{f}} i_{lq} - \frac{1}{C_{f}} i_{oq} \\ \frac{d i_{od}}{dt} = \frac{- r_{c}}{L_{C}} i_{od} + ω_{0} i_{oq} + \frac{1}{L_{C}} v_{od} - \frac{1}{L_{C}} v_{bd} \\ \frac{d i_{od}}{dt} = \frac{- r_{c}}{L_{C}} i_{oq} - ω_{0} i_{od} + \frac{1}{L_{C}} v_{oq} - \frac{1}{L_{C}} v_{bq} \end{matrix}

By combining equations (31) and (35), the state-space model of the AC MG is achieved.

Remark 5. According to equation (33), the designed controller (nominal voltage) will appear in the voltage setpoint $v_{i}^{* od}$ in the droop equation. As the output voltage $v_{i}^{od}$ of $D G_{i}$ varies, the proportionate changes appear in the controller. The change in the output current’s dynamics $i_{i}^{od}$ influences the dynamics of $v_{i}^{od}$ in equation (35). Hence, the output current deviations will affect the nominal voltage $v_{i}^{n}$ in equation (33). Furthermore, the variations in $v_{i}^{od}$ will affect the primary-layer controller and their input reference values (see equation (34)).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iDs

Hamidreza Ashrafi

Peyman Bagheri

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