Fixed-time Distributed Voltage and Reactive Power Compensation of Islanded Microgrids Using Sliding-mode and Multi-agent Consensus Design

Mohamed Ghazzali; Mohamed Haloua; Fouad Giri

网刊加载中。。。

使用Chrome浏览器效果最佳，继续浏览，你可能不会看到最佳的展示效果，

确定继续浏览么?

复制成功，请在其他浏览器进行阅读

Fixed-time Distributed Voltage and Reactive Power Compensation of Islanded Microgrids Using Sliding-mode and Multi-agent Consensus Design PDF

- ORCID：
Mohamed Ghazzali
✉
- ORCID：
Mohamed Haloua
✉
- ORCID：
Fouad Giri
✉

Electrical Engineering Department, Mohammadia School of Engineers, University Mohammed V, Rabat, Morocco； Caen Automation Laboratory (LAC), Caen Normandie University (UNICAEN), Caen, France

Updated：2022-01-21

DOI：10.35833/MPCE.2019.000308

Abstract

This paper investigates a fixed-time distributed voltage and reactive power compensation of islanded microgrids using sliding-mode and multi-agent consensus design. A distributed sliding-mode control protocol is proposed to ensure voltage regulation and reference tracking before the desired preset fixed-time despite the unknown disturbances. Accurate reactive power sharings among distributed generators are maintained. The secondary controller is synthesized without the knowledge of any parameter of the microgrid. It is implemented using a sparse one-way communication network modeled as a directed graph. A comparative simulation study is conducted to highlight the performance of the proposed control strategy in comparison with finite-time and asymptotic control systems with load power variations.

Keywords

Secondary controller; microgrids; reactive power control; voltage control

I. Introduction

MICROGRIDS are small-scale electrical distribution networks, consisting of distributed power sources, loads and energy storage systems. Primary control of microgrids maintains the stability of frequency and voltage, which causes the magnitudes of frequency and voltage to deviate from their nominal values. Emerged as a natural control system for microgrids, distributed secondary control restores the frequency and the voltage to their nominal operation points and achieves accurate power sharing in both grid-connected and islanded operation modes.

Distributed secondary control has been widely discussed in literature [

1]-[11]. Several techniques including proportional-integral (PI) control [9], feedback control [3]-[6], [10]-[12], adaptive control [6], [13] and model predictive control (MPC) [14], [15] among others are developed to address secondary control objectives of microgrids, which are voltage control, frequency control and power sharing. However, many shortcomings are to be addressed. In [9], a distributed PI controller is proposed for secondary control of voltage and frequency. Only small signal stability is guaranteed which means that large external disturbances, i.e., load variations during peak hours and the connection or disconnection of a power source, can destabilize the power system and cause voltage or frequency to swell or sag. In [5], [6], [10]-[12], feedback linearization and variants of feedback control are used to achieve secondary control of microgrids. However, a nonlinear model of AC microgrids is used in the design of the distributed control protocols that require detailed parameters of the microgrid. Practically, load parameters and transmission line impedances among other parameters are unknown and time-variant, which is challenging to apply the controllers in real applications. Adaptive control is suggested in [6], [13] for secondary control of AC microgrids. A major drawback of this approach is high computation complexity of the control law. In fact, in the adaptation technique, the control gains are updated in real time via a complicated and time-consuming mathematical approach as linear matrix inequalities (LMIs) or differential equation systems in [6] and neural networks in [13]. Thus, a control hardware with high computation capacity is required, which increases the implementation cost of the control system. In the case of neural network based adaptive control, a prior learning phase of the neurons based on experimental data is required for the design of the controller, which complicates the design, tuning and implementation of the controller. In [14], [15], the model of predictive secondary controllers is proposed for AC microgrids. The MPC approach is based on the model, which indicates that a model of the microgrid is required to calculate the output of the controller. Consequently, its robustness will depend on the accuracy of the model, which makes the controller vulnerable to modeling uncertainties. In addition, a fully detailed model of the microgrid can not be used in the design procedure as some parameters are practically unknown or difficult to obtain, which means that the unmodeled dynamics of the microgrid will always hinder the performance of the controller.

Furthermore, the distributed secondary controllers can be categorized into asymptotic controllers [

2], [5]-[10] and finite-time controllers [1], [3], [4]. In asymptotic control, the reference is only a limit of the terminal voltage or frequency of DG when time goes to infinity. Practically, reference tracking and disturbance rejection cannot be achieved in a finite time as the tracking error cannot be canceled using asymptotic control protocols. In finite-time control, the reference is reached in a finite time. The settling-time depends on the uncertain initial conditions of the system. An explicit relation between the parameters of the controller and the settling-time is not always available.

Sliding-mode control (SMC) is widely used to design the control systems and has been applied in the areas including robotics [

16]-[19], power converter control [20]-[23], motor control [24]-[27], and power systems [28]-[32]. SMC is designed to achieve reference tracking and regulation despite uncertainties of the parameter and external disturbance, which makes it suitable for applications with unknown dynamics and under uncertain conditions. In power systems, parameter uncertainties and external disturbances are constant issues [33]-[37]. In fact, most of the parameters are time-varying, which makes exact measurement of these parameters a challenging task. The fluctuations of voltage and frequency are caused by the nonlinear nature of loads and their unpredictable power consumption variations. As a result, SMC is suitable for microgrid control as it is robust to the unmodeled dynamics, variated parameters, and unknown disturbances. In addition, SMC has a straightforward design procedure and several variants of SMC have been designed with continuous and discontinuous outputs for several classes of linear and nonlinear systems. Therefore, the sliding-mode approach can be easily applied.

Considering the advantages of SMC and obviating the shortcomings of the secondary microgrid controllers, a fixed-time distributed SMC-based control approach is proposed for the secondary control of AC microgrid. Hence, the voltage regulation and reference tracking before the desired preset fixed-time can be ensured. And the accuracy of reactive power sharing is ensured among distributed generators (DGs) at its nominal levels. These objectives are guaranteed despite the uncertain microgrid parameters and the unknown disturbances. Moreover, the design procedure is model-free since no prior knowledge of load power demand, transmission line impedance or the microgrid topology is required.

The rest of the paper is organized as follows. In Section II, a large-signal dynamic model of islanded microgrids is presented. Section III presents the design of the fixed-time distributed secondary controller with voltage and reactive power sharing. A comparative simulation study on finite-time asymptotic secondary controllers and conventional power sharing protocol with load power variations is presented in Section IV. Finally, Section V concludes the paper.

II. Large-signal Dynamical Model of Islanded Microgrids

An islanded microgrid is adopted with $n$ DGs, where every DG $i$ , $i \in {1,2, . . ., n}$ contains a primary energy source connected to a voltage-source converter (VSC), an RL series filter, a step-up transformer ( $Y - Δ$ ) with transformation ratio $m_{i}$ , and a shunt capacitor $C_{t, i}$ attenuating the impact of high-frequency voltage harmonics of the local load. Each DG $i$ is connected to a set $N_{i} \subset {1,2, . . ., n}$ of neighboring DGs at the corresponding point of common coupling (PCC) through transmission lines modeled as a RL series circuits. A schematic of two connected DGs is depicted in Fig. 1, where $V_{i, d}^{*}$ and $V_{i, q}^{*}$ are the references of the direct and quadratic components of the output voltage of DG i, respectively; $V_{t, i}$ , $I_{t, i}$ , $R_{t, i}$ , $L_{t, i}$ are the voltage, current, resistance, and impetance of VSC terminal, respectively; and the subscript $i + 1$ denotes DG $i + 1$ .

Fig. 1 Schematic of two connected DGs.

DG $i$ can be modeled in the d-q framework by the follwoing large-signal dynamical model [

38]:

\{\begin{array}{l} I_{i, d} = m_{i} I_{t, i, d} - C_{t, i} \frac{d}{d t} V_{i, d} + C_{t, i} ω_{n} V_{i, q} \\ I_{i, q} = m_{i} I_{t, i, q} - C_{t, i} \frac{d}{d t} V_{i, q} - C_{t, i} ω_{n} V_{i, d} \\ \frac{d}{d t} x_{i} = A_{i} x_{i} + \sum_{j \in N_{i}} A_{i j} x_{j} + B_{i} u_{i} + D_{i} w_{i} \\ V_{t, i, d} = V_{t, i, d}^{*} \\ V_{t, i, q} = V_{t, i, q}^{*} \\ V_{t, i, d}^{*} = - ω_{n} L_{t, i} I_{t, i, q} + K_{p c i} (I_{t, i, d}^{*} - I_{t, i, d}) + K_{i c i} \int (I_{t, i, d}^{*} - I_{t, i, d}) d \\ V_{t, i, q}^{*} = ω_{n} L_{t, i} I_{t, i, d} + K_{p c i} (I_{t, i, q}^{*} - I_{t, i, q}) + K_{i c i} \int (I_{t, i, q}^{*} - I_{t, i, q}) d \\ I_{t, i, d}^{*} = F_{i} I_{i, d} - ω_{n} C_{t, i} V_{i, q} + K_{p v i} (V_{i, d}^{*} - V_{i, d}) + K_{i v i} \int (V_{i, d}^{*} - V_{i, d}) d \\ I_{t, i, q}^{*} = F_{i} I_{i, q} - ω_{n} C_{t, i} V_{i, d} + K_{p v i} (V_{i, q}^{*} - V_{i, q}) + K_{i v i} \int (V_{i, q}^{*} - V_{i, q}) d \end{array}

(1)

where $I_{i, d}$ and $I_{i, q}$ are the direct and quadratic components of the output current of DG i, respectively; $V_{i, d}$ and $V_{i, q}$ are the direct and quadratic components of the output voltage of DG i, respectively, and the output voltage of DG i is also the voltage of PCC $i$ ; $F_{i}$ is the feed-forward coefficent; $V_{t, i, d}^{*}$ and $V_{t, i, q}^{*}$ are the references of the direct and quadratic components of the terminal voltage of VSC, respectively, which are calculated by the local voltage and current of VSC controller [

39];

V_{t, i, d}

and

V_{t, i, q}

are the direct and quadratic components of VSC terminal voltage, respectively;

I_{t, i, d}^{*}

and

I_{t, i, q}^{*}

are the references of the direct and quadratic components of the VSC terminal current, respectively;

I_{t, i, d}

and

I_{t, i, q}

are the direct and quadratic components of the VSC terminal current, respectively;

I_{L, i, d}

and

I_{L, i, q}

are the direct and quadratic components of the current of load i;

ω_{n} = 2 π f_{n}

is the nominal pulsation with

f_{n} = 50

Hz nominal frequency;

A_{i}

A_{i j}

B_{i}

, and

D_{i}

are the state-space matrices [40];

x_{i} = [V_{i, d}, V_{i, q}, I_{t, i, d}, I_{t, i, q}]^{T}

is the state vector;

u_{i} = [V_{t, i, d}, V_{t, i, q}]

is the input;

K_{p v i}

K_{i v i}

K_{p c i}

, and

K_{i c i}

are the control gains of the PI controllers in the internal voltage and current control loops designed for high-frequency disturbance rejection and the filter output damping to avoid any resonance with the external network, respectively [39], which are tuned using the symmetrical optimum tool [41]; and

w_{i} = [I_{L, i, d}, I_{L, i, q}]

is the load current consumption considered as a known disturbance and an exogenous input to the system for

i \in {1,2, . . ., n}

. With the relatively high switching frequency of VSC, it is safe to neglect the switching artifact via average-value modeling. Since the dynamics of the DC-bus can be safely neglected assuming an ideal source from the DG side [39], we can obtain

V_{t, i, d} = V_{t, i, d}^{*}

and

V_{t, i, q} = V_{t, i, q}^{*}

The state-space matrices $A_{i}$ , $A_{i j}$ , $B_{i}$ , and $D_{i}$ are defined as [

40]:

A_{i} = [\begin{matrix} - \frac{1}{C_{t, i}} \sum_{j \in N_{i}} \frac{R_{i j}}{Z_{i j}^{2}} & ω_{i} - \frac{1}{C_{t, i}} \sum_{j \in N_{i}} \frac{X_{i j}}{Z_{i j}^{2}} & \frac{m_{i}}{C_{t, i}} & 0 \\ - ω_{i} + \frac{1}{C_{t, i}} \sum_{j \in N_{i}} \frac{X_{i j}}{Z_{i j}^{2}} & - \frac{1}{C_{t, i}} \sum_{j \in N_{i}} \frac{R_{i j}}{Z_{i j}^{2}} & 0 & \frac{m_{i}}{C_{t, i}} \\ - \frac{R_{t, i}}{L_{t, i}} & 0 & - \frac{m_{i}}{L_{t, i}} & ω_{i} \\ 0 & - \frac{m_{i}}{L_{t, i}} & - ω_{i} & - \frac{R_{t, i}}{L_{t, i}} \end{matrix}]

(2)

\{\begin{array}{l} A_{i j} = \frac{1}{C_{t, i}} [\begin{matrix} \frac{R_{i j}}{Z_{i j}^{2}} & \frac{X_{i j}}{Z_{i j}^{2}} & 0 & 0 \\ - \frac{X_{i j}}{Z_{i j}^{2}} & \frac{R_{i j}}{Z_{i j}^{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ B_{i} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ \frac{1}{L_{t, i}} & 0 \\ 0 & \frac{1}{L_{t, i}} \end{matrix}] \end{array}

(3)

D_{i} = [\begin{matrix} - \frac{1}{C_{t, i}} & 0 \\ 0 & - \frac{1}{C_{t, i}} \\ 0 & 0 \\ 0 & 0 \end{matrix}]

(4)

where $ω_{i} = 2 π f_{i}$ is the output pulsation of DG i and $f_{i}$ is its output frequency; $Z_{i j} = \sqrt[]{R_{i j}^{2} + ω_{i}^{2} L_{i j}^{2}}$ is the transmission line impedance; and $X_{i j} = ω_{i} L_{i j}$ is the transmission line reactance.

The output active and reactive power $P_{i}$ and $Q_{i}$ can be calculated from the output voltage and current in the d-q frame using:

\{\begin{matrix} \frac{d}{d t} P_{i} = - f_{c i} P_{i} + f_{c i} (V_{i, d} I_{i, d} + V_{i, q} I_{i, q}) \\ \frac{d}{d t} Q_{i} = - f_{c i} Q_{i} + f_{c i} (V_{i, q} I_{i, d} - V_{i, d} I_{i, q}) \end{matrix}

(5)

where $f_{c i}$ is the cut-off frequency of the low-pass filters used to extract the fundamental component of $P_{i}$ and $Q_{i}$ .

The communication network of the microgrid is described using graphs. Consider a $n$ -order weighted directed graph (digraph) $G = (V, E)$ with $V = (v_{1}, v_{2}, . . ., v_{n})$ set of nodes and $E \subseteq V \times V$ set of directed edges. $A$ weight $a_{i j} \geq 0$ is associated with every edge. $a_{i j} > 0$ if there is an edge from node $j$ to node $i$ . If agent $j$ communicates its state information to agent $i$ , then $a_{i j} = 0$ . $A = (a_{i j})_{n \times n}$ is called the adjacency matrix of the graph $G$ . The graph Laplacian matrix of $G$ , $L (G) = L$ , is defined as $L = D - A$ , where $D = d i a g {d_{1}, d_{2}, . . ., d_{n}}$ and $d_{i} = \sum_{j = 1}^{n} a_{i j}$ .

Unlike the existing models of microgrids, the interactions between DGs are considered in the proposed model. It also includes the nonlinearties introduced by the filter, the shunt capacitance, the loads, and the step-up transformers.

III. Fixed-time Distributed Secondary Voltage and Reactive Power Control

The control system developed in this paper is a secondary control strategy with voltage and reactive power sharing in hierarchical control framework. The primary control is given as:

\{\begin{array}{l} V_{i, d}^{*} = V_{i n d} - D_{q} Q_{i} \\ V_{i, q}^{*} = 0 \end{array}

(6)

where $V_{i n d}$ and $D_{q}$ are the secondary control output voltage and reactive power droop coefficient, respectively. DG $i$ generates the desired voltage reference, thus $V_{i, d}^{*} = V_{i, d}$ and $V_{i, q}^{*} = V_{i, q}$ . Then, (6) can be written as:

\{\begin{array}{l} V_{i, d} = V_{i n d} - D_{q} Q_{i} \\ V_{i, q} = 0 \end{array}

(7)

Since $V_{q i} = 0$ , the output voltage magnitude of DG i $V_{i, m a g}$ satisfies $V_{i, m a g} = \sqrt[]{V_{i, d}^{2} + V_{i, q}^{2}}$ . Thus, controlling the output voltage magnitude is the same as controlling its direct component.

The primary voltage and reactive power control aligns the output voltage magnitude to the d-axis of the voltage reference.

Secondary voltage control is designed to ensure a good trade-off between the conflicting objectives of voltage regulation and maintain the reactive power sharing accuracy in the same pattern as in the primary control:

D_{q 1} Q_{1} = D_{q 2} Q_{2} = . . . = D_{q n} Q_{n} = Δ V_{m a x}

(8)

where $Δ V_{m a x}$ is the maximum allowable voltage deviations.

Differentiating the voltage droop characteristics twice will yield:

\frac{d^{2}}{d t^{2}} V_{i n d} = \frac{d^{2}}{d t^{2}} V_{i, d} + D_{q} \frac{d^{2}}{d t^{2}} Q_{i} = u_{v Q i}

(9)

where $u_{v Q i}$ is the auxiliary control input. According to (9), the secondary voltage and reactive power control of islanded microgrids can be transformed to a leader-follower second-order consensus problem for the following linear second-order multi-agent system:

\begin{matrix} \frac{d^{2}}{d t^{2}} V_{i n d} = u_{v Q i} & i \in {1,2, . . ., n} \end{matrix}

(10)

Definition: the leader-follower second-order consensus in multi-agent systems is presented as:

\begin{matrix} \frac{d^{2}}{d t^{2}} z_{i} (t) = v_{i} (t) & i \in {1,2, . . ., n} \end{matrix}

(11)

where $z_{i}$ is the position of the $i^{t h}$ agent; and $v_{i}$ is the $i^{t h}$ agent control input.

Leader-follower second-order consensus in multi-agent system (11) is achieved under any initial conditions and $\forall i \in {1,2, . . ., n}$ . We can obtain:

\{\begin{array}{l} \underset{t \to + \infty}{l i m} | | z_{i} (t) - z_{0} (t) | | = 0 \\ \underset{t \to + \infty}{l i m} ‖\frac{d}{d t} z_{i} (t) - \frac{d}{d t} z_{0} (t)‖ = 0 \end{array}

(12)

We propose the non-singular terminal sliding-mode auxiliary control law $u_{v Q i}$ based on the voltage magnitude and the reactive power information from the neighbors of DG i:

u_{v Q i} = u_{v_{i}} + u_{Q_{i}}

(13)

u_{v_{i}} = {(\sum_{j \in N_{i}} a_{i j} + b_{i})}^{- 1} (- γ_{v_{i}} \cdot s i g n (s_{i} (e_{v_{i}}^{p}, e_{v_{i}}^{v})) + \sum_{j \in N_{i}} a_{i j} u_{j} - \frac{p_{i} - 2 q_{i}}{p_{i}} ({(\frac{π p_{i}}{4 T_{i}^{c s} q_{i}})}^{p_{i} / (p_{i} - 2 q_{i})} e_{v_{i}}^{v^{1 - [2 q_{i} / (p_{i} - 2 q_{i})]}} F_{p_{i}, q_{i}} (e_{v_{i}}^{p}) + \frac{\sqrt[]{2} ξ_{v}}{2 η_{v} T^{c r}} μ_{τ} ((e_{v_{i}}^{v})^{2 q_{i} / (p_{i} - 2 q_{i})}) (e_{v_{i}}^{v})^{- [2 q_{i} / (p_{i} - 2 q_{i})]} ((s_{i} (e_{v_{i}}^{p}, e_{v_{i}}^{v} {))}^{1 - (2 η_{v} / ξ_{v})} + n^{2 η_{v} / ξ_{v}} ((s_{i} (e_{v_{i}}^{p}, e_{v_{i}}^{v} {))}^{2})^{2 η_{v} / ξ_{v}} (s_{i} (e_{v_{i}}^{p}, e_{v_{i}}^{v} {))}^{1 - 2 η_{v} / ξ_{v}})))

(14)

u_{Q_{i}} = {(\sum_{j \in N_{i}} a_{i j} + b_{i})}^{- 1} (- γ_{Q_{i}} \cdot s i g n (s_{i} (e_{Q_{i}}^{p}, e_{Q_{i}}^{v})) + \sum_{j \in N_{i}} a_{i j} u_{j} - \frac{p_{i} - 2 q_{i}}{p_{i}} ({(\frac{π p_{i}}{4 T_{i}^{c s} q_{i}})}^{p_{i} / (p_{i} - 2 q_{i})} e_{Q_{i}}^{v^{1 - [2 q_{i} / (p_{i} - 2 q_{i})]}} F_{p_{i}, q_{i}} (e_{Q_{i}}^{p}) + \frac{\sqrt[]{2} ξ_{q}}{2 η_{q} T^{c r}} μ_{τ} ((e_{Q_{i}}^{v})^{2 q_{i} / (p_{i} - 2 q_{i})}) (e_{Q_{i}}^{v})^{- (2 q_{i} / (p_{i} - 2 q_{i}))} ((s_{i} (e_{Q_{i}}^{p}, e_{Q_{i}}^{v} {))}^{1 - (2 η_{q} / ξ_{q})} + n^{2 η / ξ} ((s_{i} (e_{Q_{i}}^{p}, e_{Q_{i}}^{v} {))}^{2})^{2 η_{q} / ξ_{q}} (s_{i} (e_{Q_{i}}^{p}, e_{Q_{i}}^{v} {))}^{1 - (2 η_{q} / ξ_{q})})))

(15)

where $e_{i}^{p} = \sum_{j \in N_{i}} a_{i j} (v_{i} - v_{j}) + b_{i} (v_{i} - v_{0})$ and $e_{i}^{v} = \sum_{j \in N_{i}} a_{i j} ({\dot{v}}_{i} - {\dot{v}}_{j}) + b_{i} ({\dot{v}}_{i} - {\dot{v}}_{0})$ are the voltage tracking errors and the derivative, respectively; $e_{Q_{i}}^{p} = \sum_{j \in N_{i}} a_{i j} (D_{Q_{i}} Q_{i} - D_{Q_{j}} Q_{j})$ and $e_{Q_{i}}^{v} = \sum_{j \in N_{i}} a_{i j} (D_{Q_{i}} {\dot{Q}}_{i} - D_{Q_{j}} {\dot{Q}}_{j})$ are the reactive power sharing errors and the derivative, respectively, and $D_{Q_{i}}$ and $D_{Q_{j}}$ are the reactive power droop coefficients; $T^{c r}$ is a time constant; $T_{i}^{c s}$ is the lowest upper-bound of the sliding-time; $b_{i} > 0$ is the pinning gain and it is non-zero only for the agents that have access to the reference voltage amplitude $v_{0}$ ; and $p_{i}$ , $q_{i}$ , $η_{v}$ , $ξ_{v}$ , $η_{q}$ , $ξ_{q}$ , $γ_{v_{i}}$ , and $γ_{Q_{i}}$ are the control parameters that verifies the following conditions:

\{\begin{array}{l} 0 < q_{i} / p_{i} < 0.25 \\ 0 < η_{v} / ξ_{v} < 0.5 \\ 0 < η_{q} / ξ_{q} < 0.5 \\ γ_{v_{i}} > 0 \\ γ_{Q_{i}} > 0 \end{array}

(16)

Denote $B = d i a g {b_{i}}_{i \in {1, 2, . . ., n}}$ , and $D_{Q_{i}}$ is the reactive power droop coefficient. $F_{p_{i}, q_{i}} (\cdot)$ and $μ_{τ} (\cdot)$ with $τ > 0$ are defined as:

\begin{array}{l} F_{p_{i}, q_{i}} (x) = (c o s h ((x^{2})^{q_{i} / p_{i}} {))}^{p_{i} / (p_{i} - 2 q_{i})} + \frac{2 q_{i} p_{i}}{p_{i} (p_{i} - 2 q_{i})} \cdot \\ (e_{i}^{p})^{2 q_{i} / p_{i}} s i n h ((x^{2})^{q_{i} / p_{i}}) (c o s h ((x^{2})^{q_{i} / p_{i}} {))}^{2 q_{i} / (p_{i} - 2 q_{i})} \end{array}

(17)

μ_{τ} (x) = \{\begin{array}{l} s i n (\frac{π x^{2}}{2 τ^{2}}) & | x | \leq τ \\ 1 & o t h e r w i s e \end{array}

(18)

$s_{i} (x, y)$ is the sliding surface defined as:

s_{i} (x, y) = y^{p_{i} / (p_{i} - 2 q_{i})} + \frac{π p_{i}}{4 T_{i}^{c s} q_{i}} x (c o s h ((x^{2})^{q_{i} / p_{i}} {))}^{p_{i} / (p_{i} - 2 q_{i})}

(19)

Figure 2 shows the block diagram of the distributed fixed-time voltage and reactive power controller.

Fig. 2 Block diagram of fixed-time distributed voltage and reactive power controller.

$T^{c r} + 2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})} / γ_{i}$ is an upper bound of the reaching time of the secondary SMC of DG i, $i \in {1,2, . . ., n}$ .

The desired prefixed upper bound of the settling time is expressed as:

T = T^{c r} + \underset{i}{m a x} (T_{i}^{c s}) + \underset{i}{m a x} (\frac{2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})}}{γ_{i}})

(20)

Theorem: if the control protocol in (15), (16), (17), and (18) is applied, then the fixed-time leader-follower consensus tracking is achieved before the prefixed settling time upper bound (20). Thus, voltage regulation and reference tracking are met in the fixed-time $T_{s e t t l i n g}$ without hindering reactive power sharing, i.e., $T_{s e t t l i n g} \leq T$ is verified.

Proof: the result in the Theorem is valid for $u_{v_{i}}$ and $u_{Q_{i}}$ , separately [

42]. However, to prove that the result is valid for the superposition of the two nonlinear control laws, i.e.,

u_{v Q i} = u_{v_{i}} + u_{Q_{i}}

, we consider the Lyapunov function

V = \sum_{i = 1}^{n} (s_{i}^{2} (e_{v_{i}}^{p}, e_{v_{i}}^{v}) + s_{i}^{2} (e_{Q_{i}}^{p}, e_{Q_{i}}^{v}))

. Using

μ_{τ} (x) + μ_{τ} (y) \leq μ_{τ} ((x + y) / 2)

with

μ_{τ} ((x + y) / 2) = μ_{τ / \sqrt[]{2}} (x + y)

and

x^{a} + y^{a} \leq {(x + y)}^{a}

\forall x, y, a \in R_{+}^{*}

, and following similar demonstration steps as in [42], it is concluded that

\underset{t \to t_{r}}{l i m} V = 0

. Then,

\underset{t \to t_{r}}{l i m} s_{i} (x, y) = 0

, where

t_{r}

is the reaching time of the sliding surface

s_{i} (x, y) = 0

. Then,

t_{r} < T^{c r} + \underset{i}{m a x} (2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})} / γ_{i})

is verified. On the sliding surface, i.e., when

s_{i} = 0

, if we denote

e_{i} \in

{e_{v_{i}}^{p}, e_{Q_{i}}^{p}, e_{v_{i}}^{v}, e_{Q_{i}}^{v}}

, the sliding motion dynamics can be described by

{\dot{e}}_{i} = - (π p_{i} / 4 q_{i} T_{i}^{c s}) (e_{i}^{p})^{1 - (2 q / p)} c o s h (((e_{i}^{p})^{2})^{q / p})

. Thus we can obtain

2 e_{i} {\dot{e}}_{i} = - (π p_{i} / 2 q_{i} T_{i}^{c s}) (e_{i}^{p})^{2 (1 - (q / p))} c o s h (((e_{i}^{p})^{2})^{q / p})

. By denoting

V_{e} = e_{i}^{2}

, the aforementioned equation can be written as

\frac{q}{p} \frac{V_{e}^{(q / p) - 1} d V_{e}}{c o s h (V_{e}^{q / p})} = - \frac{π}{2 T_{i}^{c s}} d t

, which is equivalent to

\frac{d V_{e}^{q / p}}{c o s h (V_{e}^{q / p})} = \frac{π}{2 T_{i}^{c s}} d t

. Integrating this equation from

t_{r}

t

yields

a r c s i n (t a n h ((e_{i}^{2} {(t))}^{q / p})) = - π (t - t_{r}) / 2 T_{i}^{c s} + a r c s i n (t a n h ((e_{i}^{2} (t_{r} {))}^{q / p}))

.At the settling-time

t_{s}

of the sliding motion,

e_{i} = 0

. Thus,

t_{s} = (2 T_{i}^{c s} / π) a r c s i n (t a n h ((e_{i}^{2} (t_{r} {))}^{q / p})) \leq T_{i}^{c s}

. The tracking errors

e_{v_{i}}^{p}

e_{Q_{i}}^{p}

e_{v_{i}}^{v}

e_{Q_{i}}^{v}

i \in {1,2, . . ., n}

converge to zero at

T_{s e t t l i n g}

. Then

T_{s e t t l i n g} \leq T^{c r} + \underset{i}{m a x} (T_{i}^{c s}) + \underset{i}{m a x} (2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})} / γ_{i})

is verified. Since

v_{i} - v_{0} = (L + B) e_{v_{i}}^{p}

, the reference voltage magnitude is reached at

T_{s e t t l i n g}

Remark 1: the term $2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})} / γ_{i}$ can be neglected for sufficient high values of $γ_{i}$ and low values of $τ$ , and $T = T^{c r} + \underset{i}{m a x} (T_{i}^{c s})$ is verified by the upper bound of the prefixed settling time. However, for very small values of $T^{c r} + \underset{i}{m a x} (T_{i}^{c s})$ , the effect of the term $2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})} / γ_{i}$ can be reduced but can not be negligible without hindering the controller performance as high values of $γ_{i}$ increase the chattering effect.

Remark 2: an important feature of the proposed control approach is that the design procedure is straightforward. No knowledge of the microgrid parameter is required. The desired settling-time upper bounds are specified directly in the control law, which makes the tuning process simple. And the only step left is to choose $γ_{i}$ and $τ$ as explained in Remark 2. In addition, the convergence at the desired settling-time is mathematically guaranteed despite the unknown disturbances.

IV. Comparative Simulation Study

The performance of the proposed fixed-time secondary voltage control is verified with load power variations in comparison with finite-time and asymptotic secondary voltage control. The performance of the proposed protocol of fixed-time reactive power sharing in maintaining power sharing accuracy is compared with the conventional one in [

3], [4] and others representing the benchmark for power sharing accuracy, i.e, the nominal level to be maintained within the microgrid. The test system used in the simulations is a low-voltage islanded AC microgrid (nominal frequency

f_{0} = ω_{0} / (2 π) = 50

Hz and nominal voltage magnitude

V_{0} = 380

V) containing four DGs and four loads connected via the communication network. And the network is modeled by the directed graph depicted in Fig. 3. The arrows in the graph indicates the voltage and reactive power data flow direction. DG1 is the leader node, thus the only DG gets access to the reference voltage magnitude with the pinning gain

b_{1} = 1

. For the other DGs,

b_{i} = 0

i \in {2,3, 4}

Fig. 3 Topology of microgrid.

Except for the internal voltage and current loop parameters, the DG specifications are adopted from [

40]. The filter parameters and the droop coefficients are summarized in Table I. The droop coefficients are assumed so that the voltage deviations are less than the maximum allowable voltage deviation below

10 %

[43]. Therefore, to achieve high control performance, the maximum allowable voltage deviation is

Δ V_{m a x} = 2 %

. The shunt capacitance

C_{t} = 62.86 μ H

and the transformer ratio

k = 0.6 / 13.8

are used for the four DGs. Active and reactive power variations of the loads are presented in Figs. 4 and 5. The control gains of the internal voltage and current loops are tuned as follows:

K_{p v} = K_{p v i} = 150

K_{i v} = K_{i v i} = 250

K_{p c} = K_{p c i} = 100

, and

K_{i c} = K_{i c i} = 10

\forall i \in {1,2, 3,4}

Table I Microgrid Parameters

DG	Filter parameter		Droop coefficient
DG	$R_{t} (m Ω)$	$L_{t} (μ H)$	$D_{p}$	$D_{q}$
1	1.2	93.7	2.08 $\times 10^{- 5}$	1.210 $\times$ $10^{- 3}$
2	1.6	94.8	2.10 $\times 10^{- 5}$	1.214 $\times$ $10^{- 3}$
3	1.5	107.7	2.11 $\times 10^{- 5}$	1.217 $\times$ $10^{- 3}$
4	1.5	90.6	2.12 $\times 10^{- 5}$	1.218 $\times$ $10^{- 3}$

Fig. 4 Active power variation of load.

Fig. 5 Reactive power variation of load.

The parameters of voltage and reactive power controllers are set as: $q_{i} / p_{i} = 3 / 37$ , $η_{v} / ξ_{v} = η_{q} / ξ_{q} = 0.1$ , $T^{c r} = 5 m s$ and $T_{i}^{c s} = 5 m s$ , $i \in {1,2, 3,4}$ . As $T^{c r} + T_{i}^{c s}$ is very small, several values of $τ$ and $γ_{i}$ are tested to reduce the term $2 τ^{(p_{i} - 2 q_{i}) / (2 q_{i})}$ without increasing the chattering effect. $τ = 1 \times 10^{- 3}$ , $γ_{v_{i}} = 2.4 \times 10^{6}$ , and $γ_{Q_{i}} = 10$ achieve the desired effect and will be used in this paper.

To highlight the efficiency of the proposed fixed-time control, the finite-time and the asymptotic secondary controllers in [

4] and [10] have the same settling-time as those of the proposed controller.

The simulations are conducted considering the following scenario.

1) At $t = 0$ : the simulation is initialized and the primary control is activated.

2) At $t = 0.8$ s: the proposed secondary control system is applied.

3) At $t = 1$ s: active power and reactive power of load 4 are increased.

4) At $t = 1.2$ s: active power and reactive power of load 2 are decreased.

5) At $t = 1.4$ s: load 1 is disconnected from the microgrid.

The results of the simulations are shown in Figs. 6-8. As shown in Fig. 6, the droop-based primary control maintains the microgrid voltage stability. Accurate power sharing can be shown in Fig. 8, which does not cancel the reference tracking error and maintain the microgrid back to the nominal operation conditions. Thus, the primary control results in a deviation in the voltage amplitude of DGs. As a result, the secondary control is required for voltage restoration.

Fig. 6 DG1 output voltage using proposed fixed-time control, finite-time control and asymptotic control. (a) Output voltage. (b) Zoom on time lapse [0.79 s, 0.815 s]. (c) Zoom on time lapse [0.99 s, 1.02 s]. (d) Zoom on time lapse [1.19 s, 1.22 s]. (e) Zoom on time lapse [1.39 s, 1.42 s].

Fig. 7 Control signal $V_{i n d}$ of proposed fixed-time controller, finite-time control and asymptotic control. (a) Control signal V_ind. (b) Zoom on time lapse [0.8 s, 2 s].

Fig. 8 Error of reactive power sharing using proposed and conventional approaches. (a) $D_{q 2} Q_{2} - D_{q 1} Q_{1}$ . (b) $D_{q 3} Q_{3} - D_{q 1} Q_{1}$ . (c) $D_{q 4} Q_{2} - D_{q 4} Q_{1}$ .

At $t = 0.8$ s, the secondary control is activated. Figure 6 shows that the fixed-time secondary control restores the voltage magnitude to its reference before the upper bound of prefixed settling-time of $10$ ms with less than $2$ V overshoot while maintaining smooth output voltage. However, the finite-time controller provides a fluctuating output voltage with over 12 consecutive overshoots before reaching the reference, which means that the proposed approach enhances the tracking dynamics by $90 %$ compared with that of finite-time control. Unlike the designed fixed-time approach, the asymptotic control also leads to voltage fluctuation during its convergence to the reference. And the reference tracking can not be achieved in a finite time. In the asymptotic control, the reference is only a limit of DG terminal voltage when time goes to infinity. In practical, this means that the reference tracking error can not be canceled using asymptotic control protocols, while the proposed fixed-time control ensures voltage reference tracking before a prefixed maximum settling time despite the external disturbances. For example, the reaching time of the reference is guaranteed to be finite and fixed in advance. In this paper, the fixed-time approach provides $100 %$ better performance than asymptotic approaches on both transient dynamics and steady-state basis.

The primary reactive power control represents the benchmark for power sharing accuracy, i.e., the nominal level to be maintained within the microgrid. The proposed fixed-time secondary control achieves efficient voltage reference tracking while maintaining $100 %$ of power sharing accuracy.

Load power variations begin at $t = 1$ s as shown in Figs. 4 and 5. Figure 6 shows that by using fixed-time, finite-time and asymptotic controllers, similar voltage fluctuations occur. Unlike other approaches, the proposed controller rejects the voltage fluctuations before the upper bound of prefixed settling-time of $10$ ms, which achieves accurate reactive power sharing.

The control signals displayed in Fig. 7 show that the proposed control system provides better performance with smaller control signal than finite-time and asymptotic approaches. The proposed system has faster dynamics and a smaller steady-state value, which justifies the voltage regulation and reference tracking speed of the proposed approach.

Figure 8 shows that the proposed reactive power controller maintains power sharing accuracy at its benchmark level in presence of load power variations. The voltage control has not been conducted at the expense of reactive power control. Therefore, the proposed power controller have successfully provided better voltage regulation and reference tracking performance while maintaining reactive power sharing accuracy at its nominal level.

V. Conclusion

In this paper, a fixed-time distributed voltage and reactive power secondary control approach for islanded AC microgrids has been designed. The proposed distributed sliding-mode controller ensures voltage regulation and reference tracking before the upper bound of prefixed settling-time despite the unknown disturbances. And accurate reactive power sharing among DGs is maintained. The comparative simulation conducted with load power variations confirms the performance of the controller in voltage regulation and reference tracking before the desired fixed-time, and the accuracy of reactive power sharing is maintained. Simulation results show that the proposed fixed-time control provides better performance in term of voltage regulation and reference tracking than finite-time and asymptotic approaches, which can achieve fast regulation reference tracking at the expense of the system stability with severe voltage fluctuations.

References

X. Lu, X. Yu, J. Lai et al., “A novel distributed secondary coordination control approach for islanded microgrids,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 2726-2740, Jul. 2018. [Baidu Scholar]

X. Wu, C. Shen, and R. Iravani, “A distributed, cooperative frequency and voltage control for microgrids,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 2764-2776, Jul. 2018. [Baidu Scholar]

N. M. Dehkordi, N. Sadati, and M. Hamzeh, “Distributed robust finite-time secondary voltage and frequency control of islanded microgrids,” IEEE Transactions on Power Systems, vol. 32, no. 5, pp. 3648-3659, Sept. 2017. [Baidu Scholar]

X. Wang, H. Zhang, and C. Li, “Distributed finite-time cooperative control of droop-controlled microgrids under switching topology,” IET Renewable Power Generation, vol. 11, no. 5, pp. 707-714, Feb. 2017. [Baidu Scholar]

A. Bidram, V. Nasirian, A. Davoudi et al., Cooperative Synchronization in Distributed Microgrid Control. Heidelberg: Springer International Publishing, 2017. [Baidu Scholar]

N. M. Dehkordi, N. Sadati, and M. Hamzeh, “Fully distributed cooperative secondary frequency and voltage control of islanded microgrids,” IEEE Transactions on Energy Conversion, vol. 32, no. 2, pp. 675-685, Jun. 2017. [Baidu Scholar]

J. Schiffer, T. Seel, J. Raisch et al., “Voltage stability and reactive power sharing in inverter-based microgrids with consensus-based distributed voltage control,” IEEE Transactions on Control Systems Technology, vol. 24, no. 1, pp. 96-109, Jan. 2016. [Baidu Scholar]

V. Nasirian, S. Moayedi, A. Davoudi et al., “Distributed cooperative control of DC microgrids,” IEEE Transactions on Power Electronics, vol. 30, no. 4, pp. 2288-2303, Apr. 2015. [Baidu Scholar]

J. W. Simpson-Porco, Q. Shafiee, F. Dörfler et al., “Secondary frequency and voltage control of islanded microgrids via distributed averaging,” IEEE Transactions on Industrial Electronics, vol. 62, no. 11, pp. 7025-7038, Nov. 2015. [Baidu Scholar]

A. Bidram, A. Davoudi, and F. L. Lewis, “A multiobjective distributed control framework for islanded AC microgrids,” IEEE Transactions on Industrial Informatics, vol. 10, no. 3, pp. 1785-1798, Aug. 2014. [Baidu Scholar]

A. Bidram, A. Davoudi, F. L. Lewis et al., “Distributed cooperative secondary control of microgrids using feedback linearization,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 3462-3470, Aug. 2013. [Baidu Scholar]

H. Cai, G. Hu, F. L. Lewis et al., “A distributed feedforward approach to cooperative control of AC microgrids,” IEEE Transactions on Power Systems, vol. 31, no. 5, pp. 4057-4067, Sept. 2016. [Baidu Scholar]

A. Bidram, A. Davoudi, F. L. Lewis et al., “Distributed adaptive voltage control of inverter-based microgrids,” IEEE Transactions on Energy Conversion, vol. 29, no. 4, pp. 862-872, Dec. 2014. [Baidu Scholar]

G. Lou, W. Gu, W. Sheng et al., “Distributed model predictive secondary voltage control of islanded microgrids with feedback linearization,” IEEE Access, vol. 6, pp. 50169-50178, Jan. 2018. [Baidu Scholar]

G. Lou, W. Gu, Y. Xu et al., “Distributed MPC-based secondary voltage control scheme for autonomous droop-controlled microgrids,” IEEE Transactions on Sustainable Energy, vol. 8, no. 2, pp. 792-804, Apr. 2017. [Baidu Scholar]

R. R. Nair, H. Karki, A. Shukla et al., “Faulttolerant formation control of nonholonomic robots using fast adaptive gain nonsingular terminal sliding mode control,” IEEE Systems Journal, vol. 13, no. 1, pp. 1006-1017, Mar. 2019. [Baidu Scholar]

M. Van, M. Mavrovouniotis, and S. Ge, “An adaptive backstepping nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 7, pp. 1448-1458, Jul. 2019. [Baidu Scholar]

H. Wang, Y. Pan, S. Li et al., “Robust sliding mode control for robots driven by compliant actuators,” IEEE Transactions on Control Systems Technology, vol. 27, no. 3, pp. 1259-1266, May 2019. [Baidu Scholar]

A. Ferrara and G. P. Incremona, “Design of an integral suboptimal second-order sliding mode controller for the robust motion control of robot manipulators,” IEEE Transactions on Control Systems Technology, vol. 23, no. 6, pp. 2316-2325, Nov. 2015. [Baidu Scholar]

Q. Yang, M. Saeedifard, and M. A. Perez, “Sliding mode control of the modular multilevel converter,” IEEE Transactions on Industrial Electronics, vol. 66, no. 2, pp. 887-897, Feb. 2019. [Baidu Scholar]

J. Liu, Y. Yin, W. Luo et al., “Sliding mode control of a three-phase AC/DC voltage source converter under unknown load conditions: industry applications,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 10, pp. 1771-1780, Oct. 2018. [Baidu Scholar]

W. Qi, S. Li, S Tan et al., “Parabolic-modulated slidingmode voltage control of a buck converter,” IEEE Transactions on Industrial Electronics, vol. 65, no. 1, pp. 844-854, Jan. 2018. [Baidu Scholar]

R. Ling, D. Maksimovic, and R. Leyva, “Second-order sliding-mode controlled synchronous buck DC-DC converter,” IEEE Transactions on Power Electronics, vol. 31, no. 3, pp. 2539-2549, Mar. 2016. [Baidu Scholar]

Q. Wang, H. Yu, M. Wang et al., “An improved sliding mode control using disturbance torque observer for permanent magnet synchronous motor,” IEEE Access, vol. 7, pp. 36691-36701, Feb. 2019. [Baidu Scholar]

M. A. M. Cheema, J. E. Fletcher, M. Farshadnia et al., “Sliding mode based combined speed and direct thrust force control of linear permanent magnet synchronous motors with first-order plus integral sliding condition,” IEEE Transactions on Power Electronics, vol. 34, no. 3, pp. 2526-2538, Mar. 2019. [Baidu Scholar]

X. Wang, M. Reitz, and E. E. Yaz, “Field oriented sliding mode control of surface-mounted permanent magnet ac motors: theory and applications to electrified vehicles,” IEEE Transactions on Vehicular Technology, vol. 67, no. 11, pp. 10343-10356, Nov. 2018. [Baidu Scholar]

M. A. M. Cheema, J. E. Fletcher, M. Farshadnia et al., “Combined speed and direct thrust force control of linear permanent-magnet synchronous motors with sensorless speed estimation using a sliding-mode control with integral action,” IEEE Transactions on Industrial Electronics, vol. 64, no. 5, pp. 3489-3501, May 2017. [Baidu Scholar]

R. Zhang and B. Hredzak, “Nonlinear sliding mode and distributed control of battery energy storage and photovoltaic systems in AC microgrids with communication delays,” IEEE Transactions on Industrial Informatics, vol. 15, no. 9, pp. 5149-5160, Sept. 2019. [Baidu Scholar]

Y. Huang, Z. Zhang, W. Huang et al., “DC-link voltage regulation for wind power system by complementary sliding mode control,” IEEE Access, vol. 7, pp. 22773-22780, Feb. 2019. [Baidu Scholar]

I. Villanueva, A. Rosales, P. Ponce et al., “Grid voltage oriented sliding mode control for DFIG under balanced and unbalanced grid faults,” IEEE Transactions on Sustainable Energy, vol. 9, no. 3, pp. 1090-1098, Jul. 2018. [Baidu Scholar]

A. Merabet, L. Labib, A. M. Y. M. Ghias et al., “Robust feedback linearizing control with sliding mode compensation for a grid-connected photovoltaic inverter system under unbalanced grid voltages,” IEEE Journal of Photovoltaics, vol. 7, no. 3, pp. 828-838, May 2017. [Baidu Scholar]

R. Guzman, L. G. de Vicuna, J. Morales et al., “Sliding-mode control for a three-phase unity power factor rectifier operating at fixed switching frequency,” IEEE Transactions on Power Electronics, vol. 31, no. 1, pp. 758-769, Jan. 2016. [Baidu Scholar]

S. Y. Gadelovits, D. Insepov, V. Kadirkamanathan et al., “Uncertainty and disturbance estimator based controller equipped with a multiple-time-delayed filter to improve the voltage quality of inverters,” IEEE Transactions on Industrial Electronics, vol. 66, no. 11, pp. 8947-8957, Nov. 2019. [Baidu Scholar]

M. Ghanaatian and S. Lotfifard, “Control of flywheel energy storage systems in the presence of uncertainties,” IEEE Transactions on Sustainable Energy, vol. 10, no. 1, pp. 36-45, Jan. 2019. [Baidu Scholar]

P. M. Shabestari, S. Ziaeinejad, and A. Mehrizi-Sani, “Reachability analysis for a grid-connected voltage-sourced converter (VSC),” in Proceedings of 2018 IEEE Applied Power Electronics Conference and Exposition (APEC), San Antonio, USA, Mar. 2018, pp. 2349-2354. [Baidu Scholar]

F. Liu, Y. Li, Y. Cao et al., “A two-layer active disturbance rejection controller design for load frequency control of interconnected power system,” IEEE Transactions on Power Systems, vol. 31, no. 4, pp. 3320-3321, Jul. 2016. [Baidu Scholar]

F. Farzan, M. A. Jafari, R. Masiello et al., “Toward optimal day-ahead scheduling and operation control of microgrids under uncertainty,” IEEE Transactions on Smart Grid, vol. 6, no. 2, pp. 499-507, Mar. 2015. [Baidu Scholar]

M. Ghazzali, M. Haloua, and F. Giri, “Modeling and adaptive control and power sharing in islanded AC microgrids,” International Journal of Control Automation and Systems, vol. 18, no. 5, pp. 1229-1241, May 2020. [Baidu Scholar]

N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis and testing of autonomous operation of an inverter-based microgrid,” IEEE Transactions on Power Electronics, vol. 22, no. 2, pp. 613-625, Mar. 2007. [Baidu Scholar]

M. S. Sadabadi, Q. Shafiee, and A. Karimi, “Plug-and-play voltage stabilization in inverter-interfaced microgrids via a robust control strategy,” IEEE Transactions on Control Systems Technology, vol. 25, no. 3, pp. 781-791, May 2017. [Baidu Scholar]

W. Leonhard, Control of Electrical Drives. Heidelberg: Springer-Verlag, 2001. [Baidu Scholar]

A. Khanzadeh and M. Pourgholi, “Fixed-time leader-follower consensus tracking of second-order multi-agent systems with bounded input uncertainties using non-singular terminal sliding mode technique,” IET Control Theory Applications, vol. 12, no. 5, pp. 679-686, Jan. 2018. [Baidu Scholar]

C. Moldovan, C. Damian, and O. Georgescu, “Voltage level increase in low voltage networks through reactive power compensation using capacitors,” Procedia Engineering, vol. 181, pp. 731-737, Jan. 2017. [Baidu Scholar]

Address:No.19 Chengxin Avenue, Jiangning District, Nanjing 211106, China

E-mail: mpce@alljournals.cn

Tel:86-25-81093060

Fax:86-25-81093040

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Fixed-time Distributed Voltage and Reactive Power Compensation of Islanded Microgrids Using Sliding-mode and Multi-agent Consensus Design PDF

Abstract

Keywords

I. Introduction

II. Large-signal Dynamical Model of Islanded Microgrids

III. Fixed-time Distributed Secondary Voltage and Reactive Power Control

IV. Comparative Simulation Study

V. Conclusion

References