Distributed event-triggered mechanism for secondary voltage control with microgrids

Abstract

This paper presents a secondary voltage control scheme with a distributed event-triggered mechanism for multiple distributed generators in droop-controlled microgrids. First, considering the issue of limited bandwidth of a communication network in a practical application, two types of distributed event-triggered mechanisms are proposed to reduce the information transmission pressure, while preserving the desired control performance. Then, based on the proposed triggering schemes, distributed secondary controllers are designed for distributed generators. Finally, simulation results demonstrate that by using the control strategy, the voltages of distributed generators are synchronised to their nominal values.

Keywords

Microgrid secondary voltage control distributed event-triggered

Introduction

Nowadays, to deal with the energy crisis and environmental degradation issues, a growing number of distributed generators (DGs) are integrated into microgrid systems (Che et al., 2015; Cho et al., 2011; Guerrero et al., 2013), which operate in either grid-connected or islanded mode. As in islanded mode, microgrids should maintain frequency and voltage stability autonomously. The regulation of output voltages and frequencies of DGs to their prescribed values in microgrids is called the secondary control problem (Guo et al., 2015; Peng and Zhang, 2016; Yazdanian and Mehrizi-Sani, 2014). Conventional secondary control relies on a microgrid centralized control centre which requires complex communication networks. However, due to the fact that a vast range of DGs are widely dispersed geographically, it calls for reliable communication networks and the centralized control scheme may reduce the reliability and stability of the microgrid systems Bidram et al. (2014); Weng et al. (2016). Compared to a centralized control strategy, distributed control with a sparse communication network is less sensitive to failures and improves the control performance by allowing local controllers to exchange information with their neighbours. During the past decades, some fruitful research on the distributed secondary control of microgrids has been conducted, for instance the work by Bidram et al. (2013b); Li et al. (2016b); Lu et al. (2016); Shafiee et al. (2014); Simpson-Porco et al. (2015), where the implementation of the distributed control requires local information transmission between DGs. Gaeini et al. (2017) pointed out that local controllers should exchange their information over a communication network whose structure is not necessarily the same as the physical power network. In an islanded mode of microgrids, there is no reference bus and also inertia is low. Therefore, the DGs can sense one another via the data receives from the communication network, and thus coordinate their frequency and voltage levels in a distributed manner.

Note that among these results, it is assumed that the information of each DG is continuously transmitted over the communication network. In addition, the traditional periodic sampling control scheme, in which the information is transmitted periodically among DGs, may yield conservative results since the sampling rate is selected for the worst case (Dong et al., 2017; Wen et al., 2016). To save the communication resources between DGs, it is desirable to design effective controllers that can work in event-driven environments and update their values only when needed. Therefore, the unnecessary redundant communication can be mitigated since the information transmitted is executed only when needed (Guo et al., 2014; Wang and Lemmon, 2011; Zhu and Jiang, 2015). The event-triggered idea has also been applied to reduce the traffic in a networked control system in the work by Dimarogonas et al. (2012); Peng et al. (2017); Yue et al. (2013), where an event-triggered condition is proposed to determine the information transmitting time. To avoid the above problems in practical applications, the researchers have endeavoured to solve them in recent years. For instance, for the energy dispatch issues of microgrids in the work by Li et al. (2016a), a distributed consensus-based optimization algorithm is introduced and the corresponding event-triggered optimization scheme is obtained, which could reduce the requirements of data exchange in microgrids. For load frequency control in the work by Peng et al. (2018), an adaptive event-triggering $H_{\infty}$ load frequency control scheme was investigated for multi-area power systems. Therefore, it is desirable to reduce the communication burdens between the sensors and controller nodes for a microgrid communication network, especially for secondary control problems.

This paper introduces a distributed event-triggered strategy for secondary controllers of microgrids with multiple DGs. In order to compensate for the voltage deviation caused by the primary control and restore them to their prescribed values, a kind of distributed cooperative control law for DGs is constructed with an event-triggered mechanism. The main contributions of this paper are summarized as follows.

The event-triggered mechanism is introduced in the microgrid secondary controller design. The information transmission instant of each DG is determined by the constructed event-triggered conditions. This method can reduce the number of data packet transmissions, and ease communication pressure.

Both the implementation of the secondary voltage controller and the designed event-triggered mechanism only require the neighbours’ information. This implies that the proposed secondary control strategy with a distributed architecture can avoid the drawbacks of a centralized control scheme as mentioned above.

A modified event-triggered mechanism with guaranteed uniform positive inter-event time intervals is proposed to avoid the potential intensive triggered information transmission among distributed generators, which can reduce the communication burdens further.

This paper is organized as follows: the ‘Problem formulation’ section provides the problem formulation. The secondary voltage control with two types of distributed event-triggered mechanism is presented in the ‘Preliminaries and main results’ section. An experimental validation of the proposed solution in an islanded MG with four DG units is presented in the ‘Simulation results’ section. The ‘Conclusion’ section concludes this paper.

Problem formulation

This section will study a microgrid operating in islanded mode which involves the inverter-interfaced DGs and the loads. As a primary DC source (e.g. wind system, photovoltaic (PV) array, etc.), DGs are connected by voltage source inverters, while the loads are connected through an inductor-capacitor (LC) filter and coupling inductance. To stabilize the output voltage of DGs when the microgrid (MG) switches to islanded mode, the primary control is implemented locally in the internal control loops of DGs by using the notable droop technique, which does not need any communication link. As is well known, the primary control consists of three controllers, e.g. the power, voltage and current controllers, and one can refer to the detailed descriptions of the three controllers in the work by Bidram et al. (2014); Shafiee et al. (2014).

The droop technique is a decentralized strategy, which shares active and reactive power among DGs and maintains the output levels of voltage and frequency within acceptable ranges. At the $i$ th DG terminal, desired relationships for inductive lines between reactive power and voltage can be given as follows

v_{omagi}^{*} = E_{ni} - n_{i} Q_{i}

(1)

where $E_{ni}$ are the references in primary control, $v_{omagi}^{*}$ is the output voltage magnitude reference value calculated by power controller, $n_{i}$ are the droop coefficients and $Q_{i}$ is the measured reactive power.

The magnitude of the DG output voltage $v_{omagi}$ is obtained as $v_{omagi} = \sqrt{v_{odi}^{2} + v_{oqi}^{2}}$ and the primary voltage control strategy aligns the voltage magnitude of each DG on the direct axis of its reference frame. By neglecting the dynamics of the current and voltage controller, the primary voltage control strategy is given as follows

{\begin{matrix} v_{odi} = E_{ni} - n_{i} Q_{i} \\ v_{oqi} = 0 \end{matrix}

(2)

To compensate the voltage deviation caused by the primary control, the secondary controller is applied to tune the voltage amplitude of every DG to their normal values by setting the primary references $E_{ni}$ . Therefore, the secondary control selects $E_{ni}$ such that the terminal voltage amplitude of each DG synchronizes to their nominal value, that is, $v_{odi} \to v_{ref}$ .

Preliminaries and main results

The secondary control of the microgrids is a tracking synchronization problem, where the DG units are interconnected via a communication network (Hong et al., 2006; Li et al., 2015). In the tracking synchronization problem, all agents need to be synchronized with a leader that acts as a command generator. For this purpose, each DG communicates with its neighbours and receives the information of neighbouring DGs. For saving communication resources, an event-triggered mechanism is proposed in the secondary control of microgrids. The details of a distributed secondary coordination control scheme with an event-triggered mechanism for the islanded MG is shown in Figure 1.

Figure 1.

A block diagram of the distributed secondary coordination control scheme.

Preliminaries

An undirected fixed graph $G = (V, E, A)$ is used to describe the communication network topology among all DGs. $V \overset{Δ}{=} (v_{1}, v_{2}, \dots, v_{N})$ is the node set where $v_{i}$ signifies the $i$ th DG. $E \overset{Δ}{=} {(v_{j}, v_{i}), if j \to i}$ is the edge set in which $j \to i$ represents that the information of the $j$ th DG can be transmitted to the $i$ th one. $A \overset{Δ}{=} {a_{ij}} \in R^{N \times N}$ is the weighted adjacency matrix of graph $G$ , where $a_{ij} = 1$ if and only if $(v_{j}, v_{i}) \in E$ , otherwise $a_{ij} = 0$ , and it is assumed that $a_{ii} = 0$ . The neighbouring set of node $v_{i}$ is denoted by $N_{i} \overset{Δ}{=} {v_{j} \in V | (v_{j}, v_{i}) \in E}$ and the cardinal number of $N_{i}$ is represented as $| N_{i} |$ . The term ‘undirected’ means that $(v_{i}, v_{j}) \in E$ if and only if $(v_{j}, v_{i}) \in E$ and ‘fixed’ means all elements in $E$ are constant. Defining the in-degree matrix $D \overset{Δ}{=} diag {d_{1}, \dots, d_{N}}$ where $d_{i} = \sum_{j \in N_{i}} a_{ij}$ , the Laplacian matrix of graph $G$ is given as $L = D - A$ . $L$ has all row sums equal to zero, that is $L 1_{N} = 0$ , with $1_{N}$ being the vector of ones with the length of $N$ .

In order to reduce the information transmission pressure, an event-triggered mechanism is introduced in the controller design. Each DG only transmits its own information, such as the output voltage to its neighbours at its triggering instant. This triggering instant is determined by a triggering condition which will be given in the following. Denote the triggering times of the output voltage of the $i$ th DG by $t_{k}^{i}$ for $k = 0, 1, 2, \dots$ .

Secondary voltage control with a distributed event-triggered mechanism

In this section, the secondary voltage control with a distributed event-triggered mechanism is designed such that the voltage magnitudes of DGs $v_{odi}$ is synchronized to the reference voltage $v_{ref}$ . The secondary voltage control is to choose appropriate control inputs $E_{ni}$ for the droop characteristic in equation (2). It should be noted that the dynamics of the voltage and current controller are much faster than the dynamics of the power controller (Bidram et al., 2013b; Pogaku et al., 2007). Therefore, by using equation (2) and the input–output feedback linearization method (Bidram et al., 2013a), a dynamic system for computing the control input $E_{ni}$ can be obtained as

{\overset{\cdot}{v}}_{odi} = {\overset{\cdot}{E}}_{ni} - n_{i} {\overset{\cdot}{Q}}_{i} = u_{i}

(3)

where $u_{1 i}$ is an auxiliary control. Considering the effectiveness of the event-triggered mechanism, the following distributed secondary voltage controllers of the $i$ th DG are designed as

u_{i} (t) = - k (Σ_{j \in N_{i}} a_{ij} (v_{odi} (t_{k}^{i}) - v_{odj} (t_{k' (t)}^{j})) + ε_{i 0} (v_{odi} (t_{k}^{i}) - v_{ref}))

(4)

where $t_{k' (t)}^{i}$ is the latest triggering instants corresponding to the output voltage of the $j$ th DG at instant $t$ .

Before giving the main results, we define $d_{i} (t) \overset{Δ}{=} v_{odi} (t) - v_{ref}$ , ${\hat{d}}_{i} (t) \overset{Δ}{=} v_{odi} (t_{k}^{i}) - v_{ref}, t \in [t_{k}^{i}, t_{k + 1}^{i})$ and the utilization ratio measurement errors with respect to the voltage output for the $i$ th DG as

e_{i} (t) \overset{Δ}{=} v_{odi} (t) - v_{odi} (t_{k}^{i}), t \in [t_{k}^{i}, t_{k + 1}^{i})

(5)

Then, combining the definitions yields

e_{i} (t) = d_{i} (t) - {\hat{d}}_{i} (t)

(6)

and

{\overset{\cdot}{d}}_{i} (t) = - k (\sum_{j \in N_{i}} a_{ij} ({\hat{d}}_{i} (t) - {\hat{d}}_{j} (t)) + ε_{i 0} {\hat{d}}_{i} (t))

(7)

Theorem 1. Assume that at least one $ε_{i 0}$ equals 1 for $i = 1, \dots, n$ , then under the distributed event-triggered secondary voltage controller given by equation (4) with the following event-triggered condition

| e_{i} (t) | > \frac{β h_{i} {(t)}^{2}}{\sum_{j = 1}^{n} a_{ij} | h_{j} (t) | + (ε_{i 0} + | N_{i} |) | h_{i} (t) | + ρ}

(8)

where

h_{i} (t) = \sum_{j \in N_{i}} a_{ij} (v_{odi} (t_{k}^{i}) - v_{odj} (t_{k' (t)}^{j})) + ε_{i 0} (v_{odi} (t_{k}^{i}) - v_{ref})

(9)

and $0 < β < 1$ , $ρ > 0$ is selected small enough, $ε_{i 0} = 1$ if and only if the $i$ th DG can receive the information of the leader, otherwise $ε_{i 0} = 0$ , then the output voltage magnitude of all DGs asymptotically achieve consensus, that is, the voltage magnitude $v_{odi}, i = 1, \dots, N$ synchronizes to the reference value $v_{ref}$ .

Proof. Let $d (t) = [d_{1} (t), d_{2} (t), \dots, d_{N} (t)]^{T}$ , and choose a Lyapunov function candidate as

V (t) = \frac{1}{2} (d (t)^{T} Ld (t) + \sum_{i = 1}^{n} ε_{i 0} d_{i} (t)^{2})

(10)

By the property of the Laplacian matrix $L$ described in the ‘Preliminaries’ subsection, the Lyapunov function $V (t)$ is positive. Based on equation (6), the derivative of $V$ with respect to $t$ along with the solution of the system given by equation (3) is

\begin{matrix} \overset{\cdot}{V} = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} d_{j} + ε_{i 0} d_{i}) {\overset{\cdot}{d}}_{i} \\ = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} (e_{j} + {\hat{d}}_{j}) + ε_{i 0} (e_{i} + {\hat{d}}_{i})) {\overset{\cdot}{d}}_{i} \\ = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} e_{j} + ε_{i 0} e_{i}) {\overset{\cdot}{d}}_{i} + \sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} {\hat{d}}_{j} + ε_{i 0} {\hat{d}}_{i}) {\overset{\cdot}{d}}_{i} \end{matrix}

(11)

where $l_{ij}$ is the element in the Laplacian matrix $L$ . From equations (7) and (9), we obtain

\sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} {\hat{d}}_{j} + ε_{i 0} {\hat{d}}_{i}) {\overset{\cdot}{d}}_{i} = - k \sum_{i = 1}^{n} h_{i}^{2}

(12)

On the one hand, since the graph $G$ is undirected and connected, we have

\begin{matrix} \sum_{i = 1}^{n} (\sum_{j = 1}^{n} l_{ij} e_{j} + ε_{i 0} e_{i}) {\overset{\cdot}{d}}_{i} = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} a_{ij} (e_{i} - e_{j}) + ε_{i 0} e_{i}) {\overset{\cdot}{d}}_{i} \\ \leq \sum_{i = 1}^{n} (\sum_{j = 1}^{n} a_{ij} | e_{j} | + (ε_{i 0} + | N_{i} |) | e_{i} |) | {\overset{\cdot}{d}}_{i} | \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{ij} | e_{j} | | {\overset{\cdot}{d}}_{i} | + \sum_{i = 1}^{n} (ε_{i 0} + | N_{i} |) | e_{i} | | {\overset{\cdot}{d}}_{i} | \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{ij} | e_{i} | | {\overset{\cdot}{d}}_{j} | + \sum_{i = 1}^{n} (ε_{i 0} + | N_{i} |) | e_{i} | | {\overset{\cdot}{d}}_{i} | \\ = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} a_{ij} k | h_{j} | + (ε_{i 0} + | N_{i} |) k | h_{i} |) | e_{i} | \end{matrix}

(13)

On the other hand, from the triggering condition given by equation (8), it can be derived that

| e_{i} (t) | \leq \frac{β h_{i} {(t)}^{2}}{\sum_{j = 1}^{n} a_{ij} | h_{j} (t) | + (ε_{i 0} + | N_{i} |) | h_{i} (t) | + ρ}

(14)

Combining equations (12) to (14) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq \sum_{i = 1}^{n} (\sum_{j = 1}^{n} a_{ij} k | h_{j} | + (ε_{i 0} + | N_{i} |) k | h_{i} | + ρ) | e_{i} | - k \sum_{i = 1}^{n} h_{i}^{2} \\ \leq - k (1 - β) \sum_{i = 1}^{n} h_{i}^{2} \end{matrix}

(15)

Applying LaSalle’s invariance principle, one can obtain that the solution $v_{odi}$ of the system given by equation (3) converges to the reference value $v_{ref}$ asymptotically. This completes the proof. □

Compared to the existing event-triggered schemes, the proposed event-triggered mechanism is derived based on the physical system dynamic equation (3). Besides, the left side of the triggering condition given by equation (8) is computed by the instantaneous measurements.

Remark 1. It is clear that the execution of the event-triggered mechanism only needs the discrete triggered information of its neighbours, which indicates that the information transmission pressure of the communication network can be sharply reduced compared with the conventional controllers based on continuous information transmission.

Theorem 1 implies that the distributed event-triggered controller given by equation (4) is sufficient to achieve the voltage control objective. As mentioned above, an event-triggered scheme has an advantage in mitigating unnecessary redundant communication since the information is broadcasted only when needed. However, this may result in a Zeno behaviour problem (Dimarogonas et al., 2012) since the proposed event-triggered scheme given by equation (8) is continuously monitoring. For this problem, we are seeking the lower bounds of the event time intervals $t_{k + 1}^{i} - t_{k}^{i}$ for $k = 1, 2, \dots$ .

Theorem 2. Considering the same conditions as Theorem 1, the event time intervals of the triggering conditions given by equation (8) are lower bounded as follows

t_{k + 1}^{i} - t_{k}^{i} > \frac{β h_{i}^{2} (t_{k + 1}^{i})}{M_{i} (\sum_{j = 1}^{n} a_{ij} | h_{j} (t_{k + 1}^{i}) | + (ε_{i 0} + | N_{i} |) | h_{i} (t_{k + 1}^{i}) | + ρ)}

(16)

where $M_{i}$ is a positive constant, therefore the Zeno behaviour can be avoided.

Proof. Consider the triggering condition given by equations (6) to (8), we have

\begin{matrix} \frac{d}{dt} | e_{i} (t) | \leq | \frac{d}{dt} e_{i} (t) | = | \frac{d}{dt} (v_{odi} (t) - v_{odi} (t_{k}^{i})) | \\ = | \frac{d}{dt} v_{odi} (t) | \leq k | h_{i} (t) | \end{matrix}

(17)

for $t \in [t_{k}^{i}, t_{k + 1}^{i})$ . Since the continuously differentiable functions $v_{odi} (t)$ converge to some constant values, the function $h_{i} (t)$ is upper bounded by some positive constants for each $i = 1, \dots, N$ . Hence, there are some positive constants $M_{i} \geq k | h_{i} (t) |$ such that $\frac{d}{dt} | e_{i} | \leq M_{i}$ , which implies $e_{i} (t) \leq M_{i} (t - t_{k}^{i})$ for $t \in [t_{k}^{i}, t_{k + 1}^{i})$ by using $e_{i} (t_{k}^{i}) = 0$ . According to the trigging condition given by equation (8), the next triggering time instant $t = t_{k + 1}^{i}$ for the $i$ th DG happens when

| e_{i} (t) | > \frac{β h_{i}^{2} (t)}{\sum_{j = 1}^{n} a_{ij} | h_{j} (t) | + (ε_{i 0} + | N_{i} |) | h_{i} (t) | + ρ}

which implies equation (16) holds. We analyse two cases here:

$h_{i} (t_{k + 1}^{i}) \neq 0$ . For this case, it can be obtained that the inter-event time interval $t_{k + 1}^{i} - t_{k}^{i}$ is strictly positive, which means that the $i$ th DG takes a finite time to execute the $k + 1$ th information sampling and broadcasting about the output voltage magnitude $v_{odi} (t)$ after the $k$ th one.

$h_{i} (t_{k + 1}^{i}) = 0$ . From equation (7), we obtain that $v_{od [i} (t_{k + 1}^{i})$ achieves the equilibrium point.

From a similar analysis in the work by Li et al. (2015), the event would not necessarily be triggered at $t = t_{k + 1}^{i}$ . Based on the above analysis of the two cases, we can ensure that the closed-loop system given by equation (3) does not exhibit the Zeno behaviour. □

Remark 2. As Theorem 2 implies, the event time interval of the triggering condition given by equation (8) is lower bounded away from zero, and thus, the Zeno behaviour can be avoided. However, the lower bounds will converge to zero since $h_{i} (t)$ converges to zero as time $t$ goes to infinity, and further, the event may be triggered intensive. Thus, the proposed event-triggered condition given by equation (8) would increase the amount of information transmission between the DGs and bring greater pressure to the communication network. In our previous work (Shi et al., 2017), this drawback was not solved. The solution to the problem is given in the next subsection.

Secondary voltage control with a modified distributed event-triggered mechanism

In this section, we modify the above event-triggered mechanism by incorporating a small positive constant to guarantee the uniform positive lower bounds of event time intervals, thus, avoiding transmission of information too frequently as mentioned in Remark 2. Then, the main result is given below.

Theorem 3. Assume that at least one $ε_{i 0}$ equals to 1 for $i = 1, \dots, n$ , then under the distributed event-triggered secondary voltage controller given by equation (4) with the following event-triggered condition

| e_{i} (t) | > \frac{β h_{i} {(t)}^{2} + η}{\sum_{j = 1}^{n} a_{ij} | h_{j} (t) | + (ε_{i 0} + | N_{i} |) | h_{i} (t) | + ρ}

(18)

where

h_{i} (t) = \sum_{j \in N_{i}} a_{ij} (v_{odi} (t_{k}^{i}) - v_{odj} (t_{k' (t)}^{j})) + ε_{i 0} (v_{odi} (t_{k}^{i}) - v_{ref})

(19)

and $0 < β < 1$ , $ρ > 0$ , $η > 0$ , $ε_{i 0} = 1$ if and only if the $i$ th DG can receive the information of the leader, otherwise $ε_{i 0} = 0$ , then the output voltage magnitude of all DGs asymptotically achieve bounded consensus, that is, the voltage magnitude $v_{odi}, i = 1, \dots, N$ converges into an arbitrary small neighbourhood of the reference value $v_{ref}$ .

Proof. Consider the system given by equation (3) and the distributed controller given by equation (4). By a similar analysis as the proof process of Theorem 1, the derivative of $V$ with respect to $t$ along with the solution of the system given by equation (3) under the triggering condition given by equation (18) is

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq \sum_{i = 1}^{n} (\sum_{j = 1}^{n} a_{ij} k | h_{j} | + (ε_{i 0} + | N_{i} |) k | h_{i} | + ρ) | e_{i} | - k \sum_{i = 1}^{n} h_{i}^{2} \\ \leq - k (1 - β) \sum_{i = 1}^{n} h_{i}^{2} + η \end{matrix}

(20)

Equation (20) implies that the solution ${(v_{od 1}, \dots, v_{odn})}$ converges into the set

\bar{E} \overset{Δ}{=} {(v_{od 1}, \dots, v_{odn}) | \sum_{i = 1}^{n} h_{i}^{2} \leq \frac{η}{k (1 - β)}}

(21)

Define the matrix $S \overset{Δ}{=} L + R$ where $R \overset{Δ}{=} diag {ε_{i 0}, \dots, ε_{n 0}}$ . By the assumption in Theorem 3, the matrix $S$ is positive definite (Hong et al., 2006). Then, we have $∥ S \hat{d} ∥^{2} = {\hat{d}}^{T} S^{T} S \hat{d} \geq λ_{\min} (S^{2}) ∥ \hat{d} ∥^{2}$ , where $\hat{d} = ({\hat{d}}_{1}, \dots, {\hat{d}}_{n})^{T}$ and $λ_{\min} (S^{2}) > 0$ is the minimum eigenvalue of the matrix $S^{2}$ . Due to the fact that $\sum_{i = 1}^{n} h_{i}^{2} = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} s_{ij} {\hat{d}}_{j})^{2} = ∥ S \hat{d} ∥^{2}$ , the following inequality holds in set $\bar{E}$

∥ \hat{d} ∥ \leq (\frac{η}{k (1 - β) λ_{\min} (S^{2})})^{\frac{1}{2}}

(22)

Then, according to $\sum_{j = 1}^{n} a_{ij} | h_{j} (t) | + (ε_{i 0} + | N_{i} |) | h_{i} (t) | + ρ \geq ε_{i 0} + 2 | N_{i} | + ρ$ , we can obtain that

∥ e_{i} ∥ \leq \frac{β \frac{η}{k (1 - β)} + η}{ε_{i 0} + 2 | N_{i} | + ρ} = \frac{β η + k (1 - β) η}{k (1 - β) (ε_{i 0} + 2 | N_{i} | + ρ)}

(23)

Next, the following inequality in set $\bar{E}$ is obtained by combining equations (22) and (23)

\begin{matrix} | d_{i} | = | {\hat{d}}_{i} + e_{i} | \leq | {\hat{d}}_{i} | + | e_{i} | \\ \leq (\frac{η}{k (1 - β) λ_{\min} (S^{2})})^{\frac{1}{2}} + \frac{β η + k (1 - β) η}{k (1 - β) (ε_{i 0} + 2 | N_{i} | + ρ)} \end{matrix}

(24)

Equation (24) indicates that $v_{odi} (t)$ converges into an arbitrary small neighbourhood of $v_{ref}$ since $η$ can be selected small enough.

Theorem 4. Considering the same conditions as in Theorem 3, the event time intervals of the triggering conditions given by equation (18) are lower bounded as follows

t_{k + 1}^{i} - t_{k}^{i} > \frac{1}{C_{i}} (β h_{i}^{2} (t_{k + 1}^{i}) + η)

(25)

where $C_{i}$ is some positive constant. Besides, the Zeno behaviour can be excluded.

Proof. Consider the triggering condition given by equation (18) and by a similar analysis in Theorem 2, we get

t_{k + 1}^{i} - t_{k}^{i} > \frac{β h_{i}^{2} (t_{k + 1}^{i}) + η}{{\bar{M}}_{i} (\sum_{j = 1}^{n} a_{ij} | h_{j} (t_{k + 1}^{i}) | + (ε_{i 0} + | N_{i} |) | h_{i} (t_{k + 1}^{i}) | + ρ)}

(26)

where ${\bar{M}}_{i}$ is some positive constant which satisfies $\frac{d}{dt} | e_{i} | \leq {\bar{M}}_{i}$ . Besides, as $v_{odi} (t)$ converges into some bounded sets in Theorem 3, some positive constants $C_{i}$ exist to ensure ${\bar{M}}_{i} (\sum_{j = 1}^{n} a_{ij} | h_{j} (t_{k + 1}^{i}) | + (ε_{i 0} + | N_{i} |) | h_{i} (t_{k + 1}^{i}) | + ρ) \leq C_{i}$ . Together with equation (26), it implies that equation (25) holds. Since $t_{k + 1}^{i} - t_{k}^{i} > \frac{η}{C_{i}}$ where $\frac{η}{C_{i}}$ is a constant, the Zeno behaviour can be excluded.

Remark 3. From Theorem 4, it can be obtained that the inter-event time intervals are uniform lower bounded by the positive constants which can avoid the issue of the potential intensive triggered information transmission among DGs as described in Remark 2. Although only the uniform boundedness result can be obtained under the modified event-triggered mechanism, the fluctuating of the system trajectory in a certain range around the equilibrium point is allowed in the practical power system.

Simulation results

In order to verify the effectiveness of distributed secondary controllers with event-triggered mechanisms designed in the ‘Preliminaries and main results’ section, some simulations for the microgrid test system of 311 V (per phase root mean square [RMS]), 50 Hz (314 rad/s) have been implemented through MATLAB/Simulink. The test system contains four DGs from Figure 2. The detailed parameters of the system are summarized in Table 1. And a fixed communication network in Figure 3 is considered in this paper. Besides, we select DG1 as the leader in the system. From the communication network, only DG1 can obtain the leader information, and the other DGs only exchange information with its neighbours. In the following, the proposed secondary voltage control method returns the DG voltage amplitudes of the islanded microgrid to their nominal values.

Figure 2.

The microgrid test system.

Table 1.

System parameters.

System quantities	Values	System quantities	Values
Microgrid frequency	50 Hz	DC voltage	800 V
Leader information		Line impedance
$ω_{ref}$	314 rad/s	$Z_{L 1}$	0.23 $Ω$ and 0.35e-3 H
$ν_{ref}$	311 V	$Z_{L 2}$	0.35 $Ω$ and 1.847e-3 H
		$Z_{L 3}$	0.23 $Ω$ and 0.35e-3 H
Output connector		Loads
$Z_{1} = Z_{2} = Z_{3} = Z_{4}$	0.03 $Ω$ and 0.35e-2 H	$L_{d 1}$	20 kW and 12 kVar
		$L_{d 3}$	15.3 kW and 7.6 kVar
Voltage controller		Current controller
$K_{pv 1} = K_{pv 2}$	0.1	$K_{pc 1} = K_{pc 2}$	15
$K_{pv 3} = K_{pv 4}$	0.05	$K_{pc 3} = K_{pc 4}$	10.5
$K_{iv 1} = K_{iv 2}$	420	$K_{ic 1} = K_{ic 2}$	20000
$K_{iv 3} = K_{iv 4}$	390	$K_{ic 3} = K_{ic 4}$	16000
Droop coefficients		Droop coefficients
$m_{1} = m_{2}$	9.4e-5	$n_{1} = n_{2}$	1.3e-3
$m_{3} = m_{4}$	12.5e-5	$n_{3} = n_{4}$	1.5e-3

Figure 3.

The communication digraph.

This section shows the simulation results under the distributed controllers given by equation (4) with the event-triggered mechanism given by equation (8). The parameters are set as $k = 20$ and $β = 0.5$ . It is assumed that the microgrid is islanded from the main grid at $t = 0$ . The simulation results are shown in Figures 4 and 5. Figure 4 indicates that under the proposed event-triggered mechanism, the output voltage amplitudes of all the four DGs are tuned to their references after 1.5 s. The four generators are chosen to show their broadcast periods of $d$ -axis voltage in Figure 5, where the $x$ -coordinates of crosses in this plot signify the triggering time, and the $y$ -coordinates denote the elapsed time since the last event triggering. The triggering conditions given by equation (8) for the third generator are triggered 14297 times, which indicates that the unnecessary redundant communication is mitigated in the context of ensuring the control purpose.

Figure 4.

Distributed generator (DG) voltages.

Figure 5.

Broadcast periods of the four generators with an event-triggered mechanism.

Then, with the modified event-triggered mechanism given by equation (18), the simulation result is shown in this section. The parameter in equation (18) $η = 1.2$ and the others are the same as above. Figure (6) shows that the output voltage magnitude of all distributed generators asymptotically converge into an small neighbourhood of the desired values, which indicates it achieves bounded consensus when the microgrid is switched to islanded mode. Figure (7) shows the broadcast periods of the third generator under the modified event-triggered mechanism, where triggering conditions given by equation (18) are triggered 2566 times. Compared with the result above, the number of the information transmissions is further reduced under the modified event-triggered mechanism since the uniform lower bounds of the inter-event time intervals can be obtained.

Figure 6.

Distributed generator (DG) voltages.

Figure 7.

Broadcast periods of the four generators with a modified event-triggered mechanism.

Conclusion

The issue of microgrid secondary control has been investigated in this paper. In order to reduce the information transmission pressure, two types of distributed event-triggered mechanisms are introduced in the controller design. By using the control strategy, the voltages and frequencies of distributed generators are synchronized to their nominal values, which has been demonstrated in the simulation results. Moreover, a non-ideal signal transmission, such as time delay and packet loss, particularly in a practical communication network will be considered in our future works.

Footnotes

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under (grant number 61533010 and 61503193).

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