Decentralized Robust Control of a Network of Inverter-Based Distributed Generation Systems

Abstract

This paper presents the design of decentralized robust controllers for a network of inverter-based distributed generation systems with LC filters in the scale of a nanogrid. Using overlapping decomposition, the network of inverters is clustered into several subnetworks such that all inverters within a subnetwork are strongly coupled and there is no or a weak coupling effect between any two inverters from different subnetworks. For the inverters within the same subnetwork, decentralized robust controllers are designed sequentially in the $\mu$-synthesis framework. In addition, all controllers are designed to be robust against $\pm 10\%$ variations in the LC filter parameters. To assess the performance of the proposed sequentially-designed controllers and compare it to that of the benchmark independently-designed ones, the distances between two neighboring inverters from the same and different subnetworks are considered to be 200 (m) and 800 (m), respectively. In this case, time-response and robustness analysis results illustrate the superiority of the proposed sequentially-designed controllers in the overlapping decomposition framework over the benchmark independently-designed ones. Moreover, transient overload and nonlinear load analyses demonstrate that the proposed sequentially-designed decentralized controllers are able to keep the load voltage within $\pm 10\%$ of the nominal value and the harmonic voltage distortions to less than $4\%$.

1. Introduction

Nowadays, due to environmental issues and lack of sufficient resources, more renewable energy generation systems such as wind power and photovoltaic (PV) are being utilized in power generation systems [1,2,3]. Although such distributed energy resources (DERs) have significant contributions to increased energy resources and a cleaner environment, their extensive usage gives rise to new challenges in the regulation of power systems as a consequence of the intermittent behavior of DERs [4,5]. Furthermore, conventional synchronous generators with large inertia are being replaced by DERs with small inertia (e.g., wind turbines) or even zero inertia (e.g., PV) [6], raising concerns about the stability of the power system.

All of the above-mentioned issues can become worse when DERs are grouped together in low-voltage (LV) stand-alone nanogrids (NGs), where the distances among DERs become so short that the coupling interconnections among them strengthen significantly. The short distances among DERs correspond to larger R/X ratios, which can cause serious stability issues in the entire NG if the controllers are designed improperly. This issue was introduced by the authors of [7], where two inverters with LC output filters were connected together through an interconnection line modeled by series resistance and inductance. It was demonstrated that if the decentralized controllers were designed independently such that the coupling interconnection was not considered in the design process, the entire control system became unstable as the distance between the two inverters grew shorter.

Motivated by the issue raised in [7] and based on the independent design of decentralized controllers, a new design scheme is proposed in this paper to address the problem of strongly coupled inverter-based NGs. Before delving into the details of the proposed control scheme, a literature review on the different control designs for the interconnected distributed generation systems in NGs is presented below.

In [8], the authors proposed an improved robust control scheme for multi-feeder micro-network systems by taking into account the load feeder voltage regulation and power sharing as a quadratic optimization problem. The authors of [9] presented a control strategy to enhance the dynamic response of power systems by designing an automatic regulator for converter-based DERs. The controller included a proportional-integral controller plus a washout filter in parallel. The authors of [10] presented a decentralized one degree of freedom robust control scheme to control the voltage of an uncertain inverter-interfaced AC microgrid (MG) in stand-alone mode. In a similar work [11], the authors proposed a decentralized robust control scheme to regulate the voltage of an off-grid inverter-interfaced MG. A robust controller was computed by solving a convex optimization problem. In [12], a communicationless decentralized control scheme was proposed for a PV/battery-based distributed NG, based on an adaptive I-V droop control method. A self-sustained hierarchical decentralized control technique was presented in [13] for NGs in isolated areas. In [14], an adaptive control strategy was proposed to regulate clustered NGs. Each NG included a PV system, a battery energy storage system (BESS), local loads, and a gateway unit. In [15], a control methodology for smart loads in NGs was presented to mitigate the voltage and power fluctuations caused by load changes and variations in the power generation. An inertial control scheme was presented in [16] to balance power and enhance stability in a hybrid NG.

In [17], the authors proposed a distributed direct power sliding-mode control scheme to replace the droop mechanism of each inverter in an islanded microgrid. A novel robust droop-based type-2 fuzzy logic controller was proposed in [18] to improve the stability of a cluster of NGs with constant power loads. In [19], the authors presented a data-driven decentralized reinforcement learning control structure to regulate the voltage of a distribution network with multiple MGs. In [20], the authors presented a multivariable adaptive robust control scheme for grid-forming stand-alone inverter-based MGs. In [21], a decentralized droop control scheme that utilizes the state of the battery charge to harmonize the power exchange was presented for an islanded LV NG. A nonlinear I-V droop control scheme based on the dynamic consensus algorithm was presented in [22] to balance the state of the charge of energy storage systems in NGs. The problem of high R/X in LV microgrids was studied in [23], where the effects of tie-line and droop controllers on the system were investigated. In [24], a decentralized model predictive voltage control scheme was proposed to compensate active and reactive power from PV inverters and electrical vehicles. Hybrid decentralized and distributed control schemes were presented in [25] to regulate the voltage in a distributed network of inverters. In [26], a distributed optimal control strategy was proposed for frequency regulation and economic dispatch of a power network. In [27], a distributed adaptive proportional–integral–derivative voltage controller was proposed and shown to be was robust against unknown uncertainties coming from external disturbances and parameter variations. In [28], the authors presented a technique to calculate the droop coefficient for voltage–current droop to consider the proportional power sharing among distributed generators.

Table 1. Characteristics of previous studies and reported performance compared to the results of this paper. R/X: Resistance to Reactance Ratio, $HVD$: Harmonic Voltage Distortion.

Ref.	Voltage Control Scheme	R/X	Robustness Parameters	Overload Analysis	Nonlinear Load
Afshari et al. [20]	Multivariable Adaptive Robust	0.76	—	$\Delta V<\pm 10\%$	$HVD\le 4\%$
Wang et al. [19]	MPC	—	—	$\Delta V<\pm 10\%$	—
Caiazzo et al. [27]	Adaptive PID	—	—	$\Delta V<\pm 10\%$	—
Jha et al. [28]	Adaptive	—	—	$\Delta V<\pm 10\%$	—
Khan et al. [8]	Impedance Estimator + Optimal	0.32	LC filter	$\Delta V<\pm 10\%$	—
Zhong et al. [9]	PI + Washout Filter	1	—	$\Delta V<\pm 10\%$	—
Derakhshan et al. [10]	Robust LMI-based	5.3	LC filter	—	—
Sadabadi et al. [11]	Robust LMI-based	0.01	LC filter	—	—
Alfaro et al. [17]	Sliding Mode	3.33	LC filter	—	$HVD\le 4\%$
This paper	Sequential Robust	$7.74$	LC filter	$\Delta V<\pm 10\%$	$HVD\le 4\%$

demonstrates the characteristics of previous studies and the reported performance compared to the results of this paper. Although the aforementioned studies have acceptable performance in regulating the voltage and frequency of the network of inverters, they have not completely investigated the effect of variations in the strength of coupling interconnections among the inverters due to distance changes. As the distance among inverters shortens, the dynamic couplings among them strengthen, which puts the entire NG at risk of instability, as demonstrated in [7]. This problem is not critical in the case of grid-connected systems, where the power system is dominantly stabilized by synchronous generators. However, it is of high significance in stand-alone NGs, where all the DERs are within short distances from each other, leading to increased risk of instability. Of the references in Table 1, [10,17] considered shorter distances (larger R/X) in their studies as well as parameter uncertainties. However, in [10] the authors did not take into account the effect of the overload disturbance and nonlinear load on the control performance; in [17], the authors did consider the nonlinear load analysis, but presented no overload analysis. Because isolated NGs have been growing rapidly in the past few years [29], it is highly critical that decentralized robust controllers be resilient against the coupling strength among the DERs, overload conditions, nonlinear loads, and system parameter variations.

In this study, a novel technique for designing decentralized robust controllers is presented for a small network of inverters such as in the case of a stand-alone NG. The design of decentralized controllers is carried out sequentially such that the impact of strong coupling interconnections is considered in the design process. This significantly improves the robustness of the entire NG against variations in coupling strength and in the other parameters of the system. It should be noted that the coupling interconnections appear as shared dynamics among DERs which need to be addressed during the control design process. Thus, the main contributions of this work are as follows:

• Proposing a procedure for systematic clustering of a network of inverter-based distributed generation systems (IBDGs) into subnetworks such that there are strong interconnection dynamics among the IBDGs of a subnetwork and weak interconnections across the subnetworks.
• Proposing a sequential controller design technique for the IBDGs within a subnetwork which takes into account the strong interconnection dynamics among the inverters and takes the weak cross-subnetwork dynamics as disturbances.
• Using the $\mu$-synthesis technique to make the control system robust against LC filter parameter variations.
• Conducting simulations to compare the superior performance of the proposed technique versus independently-designed benchmark controllers.
• Conducting simulations to illustrate the performance of the proposed sequential control scheme in terms of transient overload and nonlinear load analyses.

The rest of this study is organized as follows. In Section 2, the directed graph of the network of IBDGs is introduced, the dynamical equations of each of the IBDGs are formulated, and the overlapping decomposition framework is presented as a tool to expand the original state space model and to cluster the network of IBDGs into subnetworks. The sequential design of decentralized robust controllers using $\mu$-synthesis is formulated in Section 3. In Section 4, the simulation results are presented in terms of time response, transient overload, nonlinear load, and robustness analyses. Finally, our conclusions are presented in Section 5.

2. Methodology

2.1. Digraph of a Network of Inverter-Interfaced Systems

In this section, the concept of a directed graph (digraph) for a network of IBDGs including n inverters, which is used for the formulation of the state space model of the network, is introduced. The IBDGs are connected together through interconnection impedances.

A digraph $\mathcal{D}$ by definition comprises a non-empty finite set of vertices, represented by $\mathcal{V}\left(\mathcal{D}\right)$ (vertex set), and a finite set of edges, represented by $\mathcal{E}\left(\mathcal{D}\right)$, in which each edge is an ordered pair of two vertices. The vertices $v_i$ and $v_j$ are called adjacent (or neighbors) if an edge exists between them. The set of adjacent vertices of a vertex $v_i$ is represented by $\mathcal{N}\left(v_i\right)$, and its number of elements determines the degree of the vertex $v_i$, which is denoted by $deg\left(v_i\right)$. The adjacency matrix of a digraph is defined by an $n\times n$ matrix $P=\left[a_{ij}\right]$, in which $a_{ij}$ is the number of edges from $v_i$ to $v_j$ [30].

In a digraph of a network of IBDGs, each inverter is represented by a vertex. The direction of each edge, which appears as arrows in the digraph, is determined based on the choice of the reference positive direction of the current between two vertices. Therefore, the arrow from $v_i$ to $v_j$ means that the reference positive direction of current is from inverter $\#i$ to inverter $\#j$. Moreover, the quasi-adjacency matrix $S=\left[s_{ij}\right]$ is defined as follows:

$$s_{ij}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{the}\mathrm{edge}\mathrm{direction}\mathrm{is}\mathrm{from}{v}_i\mathrm{to}{v}_j\hfill \\ -1,\hfill & \mathrm{if}\mathrm{the}\mathrm{edge}\mathrm{direction}\mathrm{is}\mathrm{from}{v}_j\mathrm{to}{v}_i\hfill \\ 0,\hfill & \mathrm{if}\mathrm{no}\mathrm{edge}\mathrm{exists}\mathrm{between}{v}_i\mathrm{and}{v}_j\hfill \end{array}\right.$$

(1)

It can be shown that the matrix S has the following relationship with the matrix M:

$$S=P-P^T$$

(2)

As an example, consider the digraph of a network of four IBDGs connected together, as illustrated in Figure 1.

Based on the above-mentioned definitions, the matrices M and S for this digraph are calculated as follows:

$$P=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right],S=\left[\begin{array}{cccc}0& 1& -1& 0\\ -1& 0& 1& 0\\ 1& -1& 0& -1\\ 0& 0& 1& 0\end{array}\right]$$

The basic concepts of digraphs presented in this subsection are used in the next section to formulate the dynamical equations of a network of IBDGs in an islanded NG.

2.2. Dynamical Equations of a Network of Inverters

The electrical model of each of the inverters in this paper is illustrated in Figure 2, in which the output is connected to an LC filter and the input is connected to an energy storage system. The purpose of having LC filters is to eliminate the total harmonic distortion of the inverter output current. These passive filters are employed to curtail the amplitude of the harmonics and thereby improve the power quality [31]. In spite of the benefits of passive filters, they bring about undesired resonances, rendering the control design process more challenging [32]. Decentralized controllers aim to minimize the error between the load voltage $v_{loa{d}_i}$ and a sinusoidal reference voltage $v_{re{f}_i}$ such that the entire network is robust against the coupling strength among the inverters, parameter uncertainties of the LC filter, and the undesired disturbances coming from the load variations.

The associated dynamical equations of inverter $\#i$$(i=1,...,N)$ in Figure 2 are obtained as follows:

$$\begin{array}{cc}\hfill & i_{in{v}_i}=G_{in{v}_i}(v_{in{v}_i}-v_{loa{d}_i}),G_{in{v}_i}:=\frac{1}{L_{i}s}\hfill \\ \hfill & v_{loa{d}_i}=G_{v_{c_i}}[i_{in{v}_i}-i_{loa{d}_i}-{\displaystyle \sum _{k\in \mathcal{N}\left(i\right)}}\underset{i_{ik}}{\underbrace{s_{ik}{G}_{ik}(v_{loa{d}_i}-v_{loa{d}_k})}}]\hfill \\ \hfill & G_{v_{c_i}}:=\frac{1}{C_{i}s},G_{ik}:=\frac{1}{L_{ik}s+R_{ik}}\hfill \end{array}$$

(3)

where $L_i$ and $C_i$ represent the inductance and capacitance of the LC filter of inverter $\#i$ and $L_{ik}$ and $R_{ik}$ represent the inductance and resistance between inverters $\#i$ and $\#k$ ($i=1,\dots ,N$ and $k\in \mathcal{N}\left(i\right)$). Furthermore, $s_{ik}$ in this equation comes from Equation (1), and determines the sign of each term.

In order to formulate the state space model of inverter $\#i$, the following state space variables are defined: $x_i={\left[i_{in{v}_i}{v}_{loa{d}_i}\right]}^T$ is the state vector, $u_i$ is the $i^{th}$ control input, $d_i$ is the $i^{th}$ disturbance input (load current), and $x_{ik}=i_{ik}$ is the interconnection current between inverters $\#i$ and $\#k$. Therefore, the state space equations corresponding to Equation (3) are obtained as follows:

$$\begin{array}{c}{\dot{x}}_i=A_i{x}_i+B_i{u}_i+D_i{d}_i+{\displaystyle \sum _{k\in \mathcal{N}\left(i\right)}}{A}_{ik}{x}_{ik}\hfill \\ y=C_i{x}_i\hfill \end{array}$$

(4)

where $A_i$, $B_i$, $D_i$, $A_{ik}$, and $C_i$ are represented as follows:

$$\begin{array}{cc}\hfill & A_i=\left[\begin{array}{cc}0& -\frac{1}{L_i}\\ \frac{1}{C_i}& 0\end{array}\right],B_i=\left[\begin{array}{c}\frac{1}{L_i}\\ 0\end{array}\right],D_i=\left[\begin{array}{c}0\\ -\frac{1}{C_i}\end{array}\right]\hfill \\ \hfill & A_{ik}=\left[\begin{array}{c}0\\ -\frac{1}{C_i}\end{array}\right],C_i=\left[01\right]\hfill \end{array}$$

The closed-loop block diagram of inverter $\#i$ is obtained according to Equation (3), and is illustrated in Figure 3, where $K_i\left(s\right)$ and $M_i$ represent the decentralized controller and inverter gain for inverter $\#i$, respectively. Moreover, $u_i$ is the command signal generated by the controller $K_i\left(s\right)$. In this block diagram, $i_{loa{d}_i}$ is considered to be an external disturbance to the system.

Thus far, the dynamical model of inverter $\#i$ consisting of the LC filter and interconnection line dynamics has been formulated. It can be observed that the interconnection line dynamics $G_{ik}$ in (3) are shared between the neighboring inverters. In the next section, the state space representation of the entire network of IBDGs is obtained such that the shared dynamics (interconnection line dynamics) are considered for both neighboring inverters $\#i$ and $\#k$.

2.3. State Space Representation of the System in the Overlapping Decomposition Framework

The motivation in this section is to employ an effective strategy known as overlapping decomposition to identify the strong and weak coupling interconnections between every two IBDGs. Then, the entire network is clustered into multiple subnetworks based on these interconnection coupling strengths.

2.3.1. Overlapping Decomposition for Two Inverters

In this section, a brief introduction to overlapping decomposition is provided [33]. Two interconnected IBDGs are considered; their dynamic equations are presented in Equation (3) and their state space representation is expressed as follows (the matrices B, C, and D are not displayed for the sake of brevity):

$$\dot{x}=Ax$$

(5)

where $x={\left[i_{in{v}_1}{v}_{loa{d}_1}{i}_{12}{i}_{in{v}_2}{v}_{loa{d}_2}\right]}^T$ and the matrix A is expressed as follows:

$$\begin{array}{c}\hfill A=\left[\begin{array}{ccccc}0& -\frac{1}{L_1}& 0& 0& 0\\ \frac{1}{C_1}& 0& -\frac{1}{C_1}& 0& 0\\ 0& \frac{1}{L_{12}}& -\frac{R_{12}}{L_{12}}& \frac{1}{L_{12}}& 0\\ 0& 0& 0& 0& -\frac{1}{L_2}\\ 0& 0& \frac{1}{C_2}& \frac{1}{C_2}& 0\end{array}\right]\end{array}$$

(6)

By rearranging the five components of the state vector x into two overlapping vectors ${\overline{x}}_1={\left[i_{in{v}_1}{v}_{loa{d}_1}{i}_{12}\right]}^T$ and ${\overline{x}}_2={\left[i_{in{v}_2}{v}_{loa{d}_2}{i}_{21}\right]}^T$ with $i_{21}=-i_{12}$, and concatenating them into a new state vector $\overline{x}={\left[i_{in{v}_1}{v}_{loa{d}_1}{i}_{12}{i}_{in{v}_2}{v}_{loa{d}_2}{i}_{21}\right]}^T$, the extended state space equation is represented as follows:

$$\dot{\overline{x}}=\overline{A}\overline{x}$$

(7)

where the matrix $\overline{A}$ is expressed as follows:

$$\overline{A}=\left[\begin{array}{cccccc}0& -\frac{1}{L_1}& 0& 0& 0& 0\\ \frac{1}{C_1}& 0& -\frac{1}{C_1}& 0& 0& 0\\ 0& \frac{1}{L_{12}}& -\frac{R_{12}}{L_{12}}& 0& -\frac{1}{L_{12}}& 0\\ 0& 0& 0& 0& -\frac{1}{L_2}& 0\\ 0& 0& 0& \frac{1}{C_2}& 0& \frac{1}{C_2}\\ 0& \frac{1}{L_{12}}& 0& 0& -\frac{1}{L_{12}}& -\frac{R_{12}}{L_{12}}\end{array}\right]$$

(8)

Therefore, duplicating the shared state “$i_z$” of the original state space vector results in the extended state space model of the system represented by the matrix $\overline{A}$ in Equation (8). This extended model is generalized to the case of n inverters in the next subsection.

2.3.2. Extended State Space Model of the Network of IBDGs Using Overlapping Decomposition

Consider the network of IBDGs including n inverters in Equations (3) and (4) with the quasi-adjacency matrix defined in Equation (1). To extend the state space model of the entire network, the overall extended state vector $\overline{x}$ is defined as follows:

$$\begin{array}{cc}\hfill & \overline{x}={\left[\begin{array}{cccc}{\overline{x}}_1^T& {\overline{x}}_2^T& \cdots & {\overline{x}}_n^T\end{array}\right]}^T,\hfill \\ \hfill & {\overline{x}}_i={\left[\begin{array}{ccccccc}{x}_i^T& x_{i{k}_1}^T& x_{i{k}_2}^T& \cdots & x_{ik}^T& \cdots & x_{i{k}_{deg\left(i\right)}}^T\end{array}\right]}^T\hfill \end{array}$$

(9)

where ${\overline{x}}_i$ is the extended state vector of inverter $\#i$ and $x_i$ and $x_{ik}$ ($k\in \mathcal{N}\left(i\right)=\{k_1,k_2,...,k_{deg\left(i\right)}\}$) are defined in Equation (4). Based on (9), the extended state space model of the entire network is provided as follows:

$$\begin{array}{cc}\hfill & \dot{\overline{x}}=\overline{A}\overline{x}+\overline{B}\overline{u}+\overline{D}\overline{d},\hfill \\ \hfill & \overline{y}=\overline{C}\overline{x},\hfill \end{array}$$

(10)

where the matrices $\overline{A}$, $\overline{B}$, $\overline{C}$, and $\overline{D}$ are expressed as follows:

$$\begin{array}{cc}\hfill & \overline{A}=\left[\begin{array}{cccc}{\overline{A}}_1& {\overline{A}}_{12}& \cdots & {\overline{A}}_{1n}\\ {\overline{A}}_{21}& {\overline{A}}_2& \cdots & {\overline{A}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\overline{A}}_{n1}& {\overline{A}}_{n2}& \cdots & {\overline{A}}_n\end{array}\right],\hfill \\ \hfill & \overline{B}=diag({\overline{B}}_1,{\overline{B}}_2,\cdots ,{\overline{B}}_n),\hfill \\ \hfill & \overline{C}=diag({\overline{C}}_1,{\overline{C}}_2,\cdots ,{\overline{C}}_n),\hfill \\ \hfill & \overline{D}=diag({\overline{D}}_1,{\overline{D}}_2,\cdots ,{\overline{D}}_n)\hfill \end{array}$$

(11)

where ${\overline{A}}_i$, ${\overline{A}}_{ik}$, ${\overline{B}}_i$, ${\overline{C}}_i$, and ${\overline{D}}_i$ are formulated in Equations (12)–(14). The operator “diag” represents the block-diagonal combination of its arguments.

$${\overline{A}}_i=\left[\begin{array}{cccccccc}0& -\frac{1}{L_i}& 0& 0& \cdots & 0& \cdots & 0\\ \frac{1}{C_i}& 0& -\frac{s_{i{k}_1}}{C_i}& -\frac{s_{i{k}_2}}{C_i}& \cdots & -\frac{s_{ik}}{C_i}& \cdots & -\frac{s_{i{k}_{deg\left(i\right)}}}{C_i}\\ 0& \frac{s_{ik1}}{L_{i{k}_1}}& -\frac{R_{i{k}_1}}{L_{ik1}}& 0& \cdots & 0& \cdots & 0\\ 0& \frac{s_{ik2}}{L_{i{k}_2}}& 0& -\frac{R_{i{k}_2}}{L_{ik2}}& \cdots & 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ 0& \frac{s_{ik}}{L_{ik}}& 0& 0& \cdots & -\frac{R_{ik}}{L_{ik}}& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \frac{s_{i{k}_{deg\left(i\right)}}}{L_{i{k}_{deg\left(i\right)}}}& 0& 0& \cdots & 0& \cdots & -\frac{R_{i{k}_{deg\left(i\right)}}}{L_{i{k}_{deg\left(i\right)}}}\end{array}\right]$$

(12)

$${\overline{A}}_{ik}=\left[\begin{array}{ccccc}0& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& -\frac{s_{ik}}{L_{ik}}& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right]$$

(13)

$${\overline{B}}_i=\left[\begin{array}{c}\frac{1}{L_i}\\ 0\\ ⋮\\ 0\end{array}\right],{\overline{C}}_i={\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right]}^T,{\overline{D}}_i=\left[\begin{array}{c}0\\ -\frac{1}{C_i}\\ ⋮\\ 0\end{array}\right]$$

(14)

In the next subsection, the extended matrix $\overline{A}$ in Equation (11) is used to cluster the network of IBDGs into several subnetworks based on the relative norms of the off-diagonal blocks ${\overline{A}}_{ik}$ as compared to the diagonal blocks ${\overline{A}}_i$ in Equation (12).

2.3.3. Clustering the Network of IBDGs Using the Extended State Space Model

The relative norms of the off-diagonal blocks ${\overline{A}}_{ik}$, as compared to the diagonal blocks ${\overline{A}}_i$ in Equation (11), determine how strong the interconnections among the IBDGs are. This relative norm is defined as the norm of the off-diagonal block divided by the maximum norm of all the diagonal blocks. In this study, the ${\mathcal{H}}_{\infty }$ norm is used, which is defined as the maximum singular value of a matrix. The general rule to cluster the network of IBDGs is as follows (the parameter $ϵ$ determines the threshold of this clustering process):

• If $\frac{{\parallel {\overline{A}}_{ik}\parallel }_{\infty }}{\underset{j=1,\dots ,n}{max}{\parallel {\overline{A}}_j\parallel }_{\infty }}$ > $ϵ$, then the interconnection is considered to be strong, and inverters $\#i$ and $\#k$ are clustered into the same subnetwork.
• If $\frac{{\parallel {\overline{A}}_{ik}\parallel }_{\infty }}{\underset{j=1,\dots ,n}{max}{\parallel {\overline{A}}_j\parallel }_{\infty }}$ < $ϵ$, then the interconnection is considered to be weak, and inverters $\#i$ and $\#k$ are clustered into different subnetworks.

Based on the above rule, the entire network of IBDGs can be clustered into several subnetworks, allowing the decentralized robust controllers to be designed sequentially for each subnetwork. It should be noted that the decentralized control design is performed by using the original state space representation in Equation (4). The overlapping decomposition framework and the extended state space model in Equation (10) is only used to cluster the entire network into several subnetworks. In the next section, the proposed sequential design process of decentralized robust controllers for each subnetwork is presented.

3. Design of Robust Controllers

The clustering method in the previous section is used to divide the network of inverters into multiple subnetworks. For each subnetwork, the $\mu$-synthesis robust control technique is used in this section to sequentially design decentralized controllers. The dynamic equations of each IBDG in a subnetwork are similar to those in Equation (3). First, the parameter uncertainty model is formulated, then the design of decentralized controllers is presented using the proposed sequential methodology.

3.1. Parameter Uncertainty Modeling

The LC filter parameters are considered to be the uncertain parameters of IBDGs due to the $\pm 10\%$ variations from their nominal values [34]. Therefore, the closed-loop control system is required to be robust against these variations. In the context of $\mu$-synthesis robust control design, the parameter variations are modeled by multiplicative uncertainties as follows:

$$\begin{array}{cc}\hfill & L_i={\overline{L}}_i(1+{\mathcal{E}}_{L_i}{\delta }_{L_i})\hfill \\ \hfill & C_i={\overline{C}}_i(1+{\mathcal{E}}_{C_i}{\delta }_{C_i})\hfill \end{array}$$

(15)

where $L_i$ and $C_i$ represent the uncertain values of the LC filter parameters and ${\overline{L}}_i$ and ${\overline{C}}_i$ represent the nominal values. Moreover, “$\mathcal{E}$” represents the maximum percentage deviation of a parameter from its nominal value and “$\delta$” corresponds to the normalized uncertainty variable such that $\left|\delta \right|<1$.

The multiplicative uncertainties defined in (15) are replaced into the transfer functions $G_{in{v}_i}$ and $G_{v_{c_i}}$ in Equation (3). Then, the uncertain transfer functions are obtained using the upper linear fractional transformation operator ${\mathcal{F}}_u(.,.)$ [35] for inverter $\#i$ ($i=1,2,\dots ,n$), as follows:

$$\begin{array}{cc}\hfill & G_{in{v}_i}\left(s\right)=F_u({\overline{G}}_{in{v}_i}\left(s\right),{\delta }_{L_i})\hfill \\ \hfill & {\overline{G}}_{in{v}_i}\left(s\right)\stackrel{\Delta }{=}\frac{1}{{\overline{L}}_{is}}\left[\begin{array}{cc}-{\overline{L}}_i{\mathcal{E}}_{L_i}s& 1\\ -{\overline{L}}_i{\mathcal{E}}_{L_i}s& 1\end{array}\right]\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & G_{v_{c_i}}\left(s\right)=F_u({\overline{G}}_{v_{c_i}}\left(s\right),{\delta }_{C_i})\hfill \\ \hfill & {\overline{G}}_{v_{c_i}}\left(s\right)\stackrel{\Delta }{=}\frac{1}{{\overline{C}}_{is}}\left[\begin{array}{cc}-{\overline{C}}_i{\mathcal{E}}_{C_i}s& 1\\ -{\overline{C}}_i{\mathcal{E}}_{C_i}s& 1\end{array}\right]\hfill \end{array}$$

(17)

where the transfer functions with a “bar” notation represent their nominal equivalence.

To achieve a desired performance for the decentralized robust controllers, it is required that the following constraints be satisfied:

$$\begin{array}{cc}\hfill & H_{e_i}\left(s\right)W_{e_i}\left(s\right)<\frac{1}{{\delta }_{e_i}}\hfill \end{array}$$

(18)

$$\begin{array}{c}\hfill H_{u_i}\left(s\right)W_{u_i}\left(s\right)<\frac{1}{{\delta }_{u_i}}\end{array}$$

(19)

where $H_{e_i}\left(s\right)$ and $H_{u_i}\left(s\right)$ represent the transfer functions from $v_{re{f}_i}$ to $e_i$ and $i_{loa{d}_i}$ to $u_i$, respectively. In addition, ${\delta }_{e_i}$ and ${\delta }_{u_i}$ ($i=1,2,\dots ,n$) determine the tightness of the constraints. The two performance weight functions $W_{e_i}$ and $W_{u_i}$ respectively associated with the variables $e_i$ and $u_i$ play salient roles in the transient and steady state behavior of the time response of the closed-loop control system. Using trial and error, their parameters are selected to achieve the desired time response performance.

3.2. Sequential Design of Decentralized Controllers Using the $\mu$-Synthesis Technique

The block diagram of inverter $\#i$ is demonstrated in Figure 3 based on the system dynamical equations in (3) and (4) and the uncertainty and performance equations in (16)–(19). The relationships between the inputs and outputs of the $\delta$ parameters (${\delta }_{L_i}$, ${\delta }_{C_i}$, ${\delta }_{u_i}$, and ${\delta }_{e_i}$) are represented as follows:

$$I_i=\left[\begin{array}{c}{p}_{L_i}\\ p_{C_i}\\ i_{loa{d}_i}\\ v_{re{f}_i}\end{array}\right]=\underset{{\Delta }_i}{\underbrace{\left[\begin{array}{cccc}{\delta }_{L_i}& 0& 0& 0\\ 0& {\delta }_{C_i}& 0& 0\\ 0& 0& {\delta }_{u_i}& 0\\ 0& 0& 0& {\delta }_{e_i}\end{array}\right]}}\underset{O_i}{\underbrace{\left[\begin{array}{c}{q}_{L_i}\\ q_{C_i}\\ u_i\\ e_i\end{array}\right]}}={\Delta }_i{O}_i$$

(20)

where all the variables are indicated in Figure 3.

For a subnetwork of IBDGs including m inverters, the extended inverter block diagrams in Figure 3 are concatenated to form the overall subnetwork block diagram in the form of $\Delta /G/K$ (uncertainty/nominal system/controller) illustrated in Figure 4c, with the ${\Delta }_i$s defined in Equation (20).

The decentralized robust controllers $C_i(i=1,\dots ,m)$ are designed based on the robust control condition, as follows [36]:

$$\begin{array}{cc}\hfill & {\mu }_{\Delta }\left({\mathcal{F}}_L(G\left(s\right),K\left(s\right))\right)<1\hfill \\ \hfill & K\left(s\right)\stackrel{\Delta }{=}diag(K_1\left(s\right),C_2\left(s\right),\dots ,K_m\left(s\right))\hfill \end{array}$$

(21)

where $F_L(.,.)$ represents the lower LFT of its arguments and “$\mu$” denotes the $\mu$ norm [37].

The proposed sequential design procedure of decentralized robust controllers for a subnetwork of m inverters includes the following steps:

• Step 1. While controllers $K_2$, …, $K_m$, and uncertainty matrices ${\Delta }_2$,…, ${\Delta }_m$ are left open (Figure 4a), controller $K_1$ is designed for system G and uncertainty matrix ${\Delta }_1$ using the D–K iteration algorithm [36,38]. Then, controller $K_1$ is placed in the loop with system G to form the closed-loop system $G_1$.
• Step 2. While controllers $K_3$,…, $K_m$, and uncertainty matrices ${\Delta }_3$,…, ${\Delta }_m$ are left open (Figure 4b), controller $K_2$ is designed for system $G_1$ and uncertainty matrices ${\Delta }_1$ and ${\Delta }_2$ using the D–K iteration algorithm. Then, controller $K_2$ is placed in the loop with system $G_1$ to form the closed-loop system $G_2$.
  ⋮
• Step m. Controller $K_m$ is designed for system $G_{m-1}$ and the uncertainty matrices ${\Delta }_1$,…, ${\Delta }_m$ using the D–K iteration algorithm. Then, controller $K_m$ is placed in the loop with system $G_{m-1}$ to form the overall closed-loop system $G_m$, which is simulated.

The D–K iteration algorithm is illustrated in Algorithm 1 and implemented in the MATLAB environment using the command “dkit”. In this algorithm, $D_l\left(s\right)$ and $D_r\left(s\right)$ respectively represent the left and right scaling matrices, defined as follows:

$$D_l\left(s\right)=D_r\left(s\right)=diag(d_1\left(s\right),\dots ,d_4\left(s\right))$$

(22)

where $d_1\left(s\right),\dots ,d_4\left(s\right)$ respectively denote the scaling functions corresponding to the four uncertainty parameters ${\delta }_{L_i}$, ${\delta }_{C_i}$,${\delta }_{u_i}$, and ${\delta }_{e_i}$ in (20). The matrices $D_l\left(s\right)$ and $D_r\left(s\right)$ are employed to calculate an upper bound for ${\mu }_{\Delta }\left({\mathcal{F}}_L(G\left(s\right),K\left(s\right))\right)$ in (21). Interested readers are referred to [36,38], which elucidate more details about this algorithm.

Algorithm 1 D–K iteration algorithm to minimize the upper bound in (21).

1: Using the ${\mathcal{H}}_{\infty }$ control design technique, initialize the controller $K\left(s\right)$. 2: Compute the lower LFT, ${\mathcal{F}}_L(G\left(s\right),K\left(s\right))$ 3: Calculate the desired frequency range for the control loop, and select N frequencies ${\omega }_n$, $n=1,\dots ,N$, which are uniformly distributed within the frequency range. 4: Find the optimum scaling functions ${\widehat{d}}_k\left({\omega }_n\right)$, $k=1,\dots ,4$ at each ${\omega }_n$, to replace for $d_k\left(s\right)$ in (22) and build the left and right scaling matrices ${\widehat{D}}_l\left({\omega }_k\right)$ and ${\widehat{D}}_r\left({\omega }_k\right)$ to minimize ${\sigma }_{max}\left({\widehat{D}}_l\left({\omega }_n\right){\mathcal{F}}_L(G\left(j{\omega }_n\right),K\left(j{\omega }_n\right)){\widehat{D}}_r^{-1}\left({\omega }_n\right)\right)$. 5: Estimate $d_k\left(s\right)$ such that $d_k\left(s\right),d_k^{-1}\left(s\right)\in \mathcal{R}{\mathcal{H}}_{\infty }$, and $|d_k\left(j\omega \right)|\approx {\widehat{d}}_k\left({\omega }_n\right)$ $\forall {\omega }_n$. $\mathcal{R}{\mathcal{H}}_{\infty }$ denotes the all the stable real transfer functions with poles on the imaginary axis. 6: For the following system, design a ${\mathcal{H}}_{\infty }$ controller: $$\left[\begin{array}{cc}{D}_l\left(s\right)& 0\\ 0& 1\end{array}\right]{\left[\begin{array}{cc}{D}_l\left(s\right)& 0\\ 0& 1\end{array}\right]}^{-1}$$ 7: Stop if $D_l\left(s\right)$ and $D_r\left(s\right)$ are close to their estimates in the previous step. Otherwise, go to step 2.

In the next section, the proposed clustering method in the overlapping decomposition framework is applied to a network of six IBDGs and the proposed sequential design procedure is applied to each clustered subnetwork. The superiority of the sequentially designed decentralized controllers in comparison to independently designed controllers is demonstrated by simulation results. The reason for choosing six IBDGs in this study is to provide an example of a small-scale nanogrid with different interconnection lengths. This network is used to show the acceptable performance of the proposed sequential robust control design. The proposed control design procedure is general and can be applied to any number of IBDGs.

Algorithm 2 Proposed sequential control design
1: start procedure
2: $i=1$
3: while $i\le m$ do
4:  Initialize $W_{e_i}\left(s\right)$ and $W_{u_i}\left(s\right)$
5:  do
6:    $D-K$ iteration
7:    Update $W_{e_i}$ and $W_{u_i}$
8:  while ${\mu }_{diag({\Delta }_1,\dots ,{\Delta }_i)}\left({\mathcal{F}}_L(G_{i-1}\left(s\right),K_i\left(s\right))\right)\ge 1$
9:  $K_i$ order reduction
10:   i ← i + 1
11: return controllers $K_1,\dots ,K_m$
12: end procedure

4. Simulation Results

In this section, a network of IBDGs including six inverters is considered, as illustrated in Figure 5. The six inverters are connected to each other through interconnection lines indicated by impedances $Z_{ij}$. Based on Section 2.3.3, this network is clustered into three subnetworks, which are identified by three dashed circles. As indicated in Figure 5, the distances between two neighboring inverters from the same and different subnetworks are $200\left(\mathrm{m}\right)$ and $800\left(\mathrm{m}\right)$, respectively. The parameter $ϵ$ mentioned in Subsection IV-C is chosen to be 0.01. The sequential control design procedure is separatelyapplied to each of the three subnetworks. MATLAB R2021a software was used in this paper to design the robust controllers using the robust control toolbox, and the Simulink/Simscape environment with a sampling time of $100\left(\mathrm{ns}\right)$ was used to simulate the network of IBDGs shown in Figure 5. The parameter values of each of the generation systems are listed in

Table 2. Values of the system parameters.

Inverter 1	$M_{in{v}_1}=1$, $L_1=3.0\left(\mathrm{mH}\right)$, $C_1=2.0\left(\mathsf{\mu }\mathsf{F}\right)$, $L_{loa{d}_1}=1.0\left(\mathsf{H}\right)$, $R_{\mathit{l}oa{d}_1}=630\left(\mathsf{\Omega }\right)$
Inverter 2	$M_{in{v}_2}=1$, $L_2=2.8\left(\mathrm{mH}\right)$, $C_2=2.2\left(\mathsf{\mu }\mathrm{F}\right)$, $L_{loa{d}_2}=1.2\left(\mathrm{H}\right)$, $R_{loa{d}_2}=632\left(\mathsf{\Omega }\right)$
Inverter 3	$M_{in{v}_3}=1$, $L_3=2.9\left(\mathrm{mH}\right)$, $C_3=2.1\left(\mathsf{\mu }\mathrm{F}\right)$, $L_{loa{d}_3}=0.9\left(\mathrm{H}\right)$, $R_{loa{d}_3}=628\left(\mathsf{\Omega }\right)$
Inverter 4	$M_{in{v}_4}=1$, $L_4=3.2\left(\mathrm{mH}\right)$, $C_4=2.3\left(\mathsf{\mu }\mathrm{F}\right)$, $L_{loa{d}_4}=1.3\left(\mathrm{H}\right)$, $R_{loa{d}_4}=635\left(\mathsf{\Omega }\right)$
Inverter 5	$M_{in{v}_5}=1$, $L_5=3.1\left(\mathrm{mH}\right)$, $C_5=2.2\left(\mathsf{\mu }\mathrm{F}\right)$, $L_{loa{d}_5}=1.1\left(\mathrm{H}\right)$, $R_{loa{d}_5}=631\left(\mathsf{\Omega }\right)$
Inverter 6	$M_{in{v}_6}=1$, $L_6=2.7\left(\mathrm{mH}\right)$, $C_6=1.9\left(\mathsf{\mu }\mathrm{F}\right)$, $L_{loa{d}_6}=1.4\left(\mathrm{H}\right)$, $R_{loa{d}_6}=634\left(\mathsf{\Omega }\right)$
Interconnection Lines	$R_L=0.642(\mathsf{\Omega }/\mathrm{km})$, $L_L=2.2\times {10}^{-4}\left(\mathrm{H}/\mathrm{km}\right)$ [39]

. The LC filter parameters are chosen such that the total harmonic distortion of the inverter output voltage becomes minimized [7]. Furthermore, $M_{in{v}_i}$ represents the inverter gain in Figure 3.

4.1. Performance of the Proposed Control Scheme and Independent Control Design

The interconnection impedances, which consist of inductive and resistive components, are proportional to the distances among the inverters. They play an essential role in the strength of the interconnections among the inverters, and as a result in the transient and steady state behavior of the entire system. Figure 6 illustrates the time response of the proposed sequentially-designed decentralized controllers for inverter $\#4$ ($In{v}_4$). It is clear that the sequentially-designed controllers accurately follow their reference voltages. On the other hand, Figure 7 demonstrates that the independently-designed controllers (benchmark [35]) are not capable of damping the undesired oscillations caused by the strong coupling interconnections among the inverters. It should be noted that the results for the other five inverters in Figure 5 were similar to those of inverter $\#4$; hence, they are not shown here.

In the following subsections, more case studies are presented to demonstrate the acceptable performance of the proposed decentralized robust controllers in terms of transient overload, nonlinear load, and robustness.

4.2. Transient Overload Analysis

This subsection presents the impact of the peak load demand on the performance of the system in Figure 5. According to [40], the active and reactive powers of a residential load in the worse case scenario are computed to be around $21\left(\mathrm{kW}\right)$ and $10\left(\mathrm{kVAR}\right)$, respectively. To assess the performance of the proposed control scheme, the above load is applied to the NG at $0.054\left(\mathrm{s}\right)$. Figure 8 demonstrates that the proposed control scheme is able to damp the undesired oscillation caused by the peak load disturbance. Moreover, Figure 8b illustrates that the load voltage deviation from the reference voltage is within $\pm 10\%$ of the nominal value, which satisfies the IEEE 1250 standard [41].

4.3. Non-Linear Load Analysis

Due to the drastic growth of power electronic equipment such as uninterrupted power supply, rectifiers, etc., nonlinear loads have become prevalent in the modern power system industry. These loads inject unwanted harmonics into the current and voltage waveforms, resulting in lower quality electrical power [42]. In this article, the nonlinear load is modeled as a diode rectifier, which is a common model in power system studies [43,44]. Applying this nonlinear load distorts the load current and voltage. Figure 9 demonstrates the harmonic contents of the load voltage. It can be observed that the proposed sequentially-designed decentralized controllers maintain harmonic voltage distortions below the limit specified by the IEEE 1547 standard [45], according to which the maximum harmonic voltage distortion must be $4\%$ for harmonics less than 11.

4.4. Robustness Analysis

As mentioned in Section 3.1, the LC filter has uncertain parameters which are considered in the process of the robust control design. The variations of these parameters are in the range of $\pm 10\%$. Figure 10 demonstrates the time response of the sequentially-designed control system. It can be clearly seen that the voltage closely tracks the reference voltage in spite of the $10\%$ parameter variations in the closed-loop system.

5. Conclusions

This paper presents a sequential design procedure for decentralized robust control of a network of IBDGs which are connected together through inductive and resistive interconnections. Using the overlapping decomposition method, the state space model of the network is expanded with respect to the shared dynamics among the inverters. Then, the network is clustered based on the norms of the off-diagonal blocks of the extended system matrix $\overline{A}$, which are proportional to the distances among the inverters as compared to the norms of diagonal blocks. The proposed sequential $\mu$-synthesis robust controller design process is then utilized for each subnetwork (cluster). Moreover, $\pm 10\%$ variations are taken into account for the LC filter parameters of all the IBDGs in the design process of the robust controllers. The time response analysis illustrates that the proposed sequentially-designed decentralized controllers outperform the independently-designed (benchmark) ones, even under strong interconnections among the IBDGs of each subnetwork. The main reason for this is that the influence of interconnections are taken into account in the sequential design process, while they are taken as external disturbances in the independent design scheme. Moreover, transient overload and nonlinear load analyses illustrate that the proposed sequentially-designed decentralized controllers are able to keep the load voltage within $\pm 10\%$ of the nominal value (IEEE 1250 standard) and the harmonic voltage distortions to less than $4\%$ (IEEE 1547 standard). This study recommends that the proposed sequentially-designed decentralized controllers provide a valuable solution to stabilize a network of IBDGs when their distances become short, as is the case in nanogrids. However, the sequential design of decentralized controllers becomes more challenging in cases where there are many generation systems in a subnetwork. Moreover, this case requires plug-and-play operation of the system, which the proposed sequentially-designed decentralized controllers are not able to provide. Therefore, potential future work could involve the independent design of decentralized robust controllers such that their plug-and-play capability is preserved.