Distributed Control of Inverter-Based Lossy Microgrids for Power Sharing and Frequency Regulation Under Voltage Constraints

# Distributed Control of Inverter-Based Lossy Microgrids for Power Sharing and Frequency Regulation Under Voltage Constraints

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###### Abstract

This paper presents a new distributed control framework to coordinate inverter-interfaced distributed energy resources (DERs) in island microgrids. We show that under bounded load uncertainties, the proposed control method can steer the microgrid to a desired steady state with synchronized inverter frequency across the network and proportional sharing of both active and reactive powers among the inverters. We also show that such convergence can be achieved while respecting constraints on voltage magnitude and branch angle differences. The controller is robust under various contingency scenarios, including loss of communication links and failures of DERs. The proposed controller is applicable to lossy mesh microgrids with heterogeneous R/X distribution lines and reasonable parameter variations. Simulations based on various microgrid operation scenarios are also provided to show the effectiveness of the proposed control method.

USA]Chin-Yao Changand USA]Wei Zhang

Department of Electrical and Computer Engineering, Ohio State University, Columbus, OH 43210, USA

Key words:  Microgrid Control, Droop Control, Frequency Synchronization, Power Sharing, Voltage Regulation

## 1 Introduction

Microgrids are low voltage power networks comprised of distributed generations (DGs), energy storages systems (ESSs), and loads that can operate in either grid-connected or island mode. Distributed generation contributes on-site and clean energy, which is expected to make power networks more robust, efficient and environmentally friendly [1, 2]. Energy storage systems are considered as an important resource to benefit the power networks by smoothing real time imbalance between generation and demand [3]. Some storage devices such as freewheel and battery packs can be integrated with intermittent DGs to regulate the power injection to a power network [4, 5]. Demand side appliances such as plug-in hybrid electric vehicles (PHEV) and thermostatically controlled loads (TCLs) can also be viewed as energy storage resources. Those “storage” appliances can be coordinated to provide ancillary services to the main grid [6, 7, 8]. The proximity of DGs and ESSs to loads in a microgrid allows for a transition to the island mode during faults on the main grid. Such a transition may also be triggered by efficiency or reliability incentives, (see [9, 10]).

Distributed energy resources (DERs) such as DGs and ESSs connect to the microgrid through DC/AC or AC/AC inverters. During the island mode, the inverters are typically operated as voltage source inverters (VSIs). These VSIs need to be controlled cooperatively to achieve desired performance and reliability properties. In AC networks, voltage magnitude and angle difference between connected buses should be regulated in some bounded ranges for system security and stability. Frequency synchronization to a nominal value is also crucial for grid connection and stability purposes. Besides frequency and voltage regulation, sharing of active and reactive power is also considered as important control objectives in microgrids [11, 12]. They require that the power injection into the microgrid from DERs is proportional to the nominal value defined by economics or other incentives, while satisfying load demands [13]. Power sharing enables effective utilization of limited generation resources and prevents overloading [14].

To achieve the aforementioned objectives, a microgrid is typically controlled using a hierarchal structure including primary, secondary, and tertiary controls [15, 16, 17, 18], which is similar to the one used in the traditional power systems. The primary droop control of a microgrid maintains the voltage and frequency stability while balancing the generation and load with proper power sharing. The secondary controller compensates the voltage and frequency deviations from their reference values. The tertiary control establishes the optimal power sharing between inverters in both islanding and grid-connected modes.

The primary droop is generally a decentralized controller that adjusts the voltage frequency and magnitude of each inverter in response to active and reactive power deviations from their nominal values. Various droop methods are proposed to achieve proportional active and reactive load power sharing [11, 19, 20, 21, 22, 23, 24]. However, this is often achieved at the cost of sacrificing other control objectives such as voltage and frequency regulation. The secondary control utilizes either centralized or decentralized communication infrastructures to restore frequency and voltage deviation induced by the primary droop. Most of the existing secondary control methods require centralized communications [25, 26, 27]. On the other hand, decentralized secondary control has recently been proposed to avoid single point of failure [28]. The combined operations of the primary and secondary control require separation of time scale, resulting in slow dynamics that cannot effectively handle fast-varying loads [29]. In addition, the secondary control may destroy the proportional power sharing established in the primary control layer [30]. One possible solution is to adopt distributed or decentralized control structure for primary and secondary control layers to improve performance and support plug-and-play operation of the microgrid [18].

Many existing primary and secondary control methods rely on small signal linearization for stability analysis, which is vulnerable to parameter variations and change of operating points. Only several recent works [30, 13, 31] have rigorously analyzed the stability of microgrid with droop-controlled inverters. In particular, [30] derives a necessary and sufficient condition for the stability under primary droop control. The authors have also proposed a distributed averaging controller to fix the time scale separation issue between the primary and secondary control layers. In [13] and [31], stability conditions of lossless mesh microgrids have been provided. Despite their advantages, these nonlinear methods still suffer from several common limitations. First, all the nonlinear analyses mentioned above only focus on lossless microgrids with purely inductive distribution lines. The results may not be applicable for microgrids with heterogeneous and mixed R/X ratio lines, which is common in low voltage microgrids [25]. Secondly, since only frequency droop is carefully analyzed, reactive power sharing is often not guaranteed.

To address the aforementioned limitations of the existing works, we propose a distributed control framework to coordinate VSIs in an island AC microgrid. The proposed control adjusts each inverter frequency and voltage magnitude based on the active/reactive power measurements of its neighbors. We first show that the particular control structure ensures that any equilibrium of the closed-loop system results in the desired power sharing and frequency synchronization. Secondly, conditions for power sharing and synchronized frequency respecting voltage constraints are provided. The proposed controller can be applied to both radial and mesh microgrids with mixed R/X ratios. Furthermore, the proposed controller requires no separation of time scale and can tolerate reasonable parameter variations. To the authors’ knowledge, most existing control framework cannot achieves active/reactive power sharing while respecting voltage and frequency regulation for a mesh micogrid with mixed R/X ratio lines.

To demonstrate the robustness of the proposed distributed controller, we also study the control performance under partial communication failures and the plug-and-play operations. We will show that as long as the communication network remains connected, all the desired properties including power sharing and frequency and voltage regulation still hold in these contingency scenarios. This effectively demonstrates the robustness of the proposed distributed controller. It is worth to mention that the proposed framework may require faster communications among the VSIs than the traditional secondary control. However, such communication requirement is reasonable for most microgrid control systems [32, 33, 34].

The rest of this paper is organized as follows. Section 2 formulates the microgrid control problem. Sufficient conditions for the solvability of the proportional power sharing problem respecting voltage constraints are also provided. The proposed distributed control framework is developed in Section 3. Robustness of the distributed controller under loss of communication links or failures of DERs is studied in Section 4. In Section 5, we validate the proposed controller through simulations under various microgrid operating scenarios, including abrupt changes of loads and loss of one VSI. Some concluding remarks are given in Section 6.

Notation Define and as positive and negative real numbers, respectively. Denote . Given a set , let and be its cardinality and power set, respectively. Denote the diagonal matrix of a vector as . For a set of vectors , , let be the augmented vector of collecting all . Given a polyhedron , let be the vertex set of . For a closed set , int() and are the interior and the boundary of . The distance between a point and the set is denoted as . Define and as the vectors with all the elements being ones and zeros, respectively. For a symmetric matrix , let and be the spectrum and minimal eigenvalue of , respectively. Denote as the tensor product between matrices and . Let null() be the null space a matrix .

## 2 Problem Formulation

In this paper, we consider a connected island microgrid network as shown in Fig. 1. An island microgrid is represented by a connected and undirected graph , where is the set of buses (nodes) and is the set of distribution lines (edges) connecting the buses. The set of buses is partitioned into two parts, inverter buses and load buses . Let , and . The magnitude and phase angle of the bus voltage are denoted as and , respectively. Let be the state vector at bus , and let and be the inverter bus state vector and load bus state vector, respectively. The overall system state vector is denoted by and will be referred to as the system voltage profile.

For each bus , let and be the active and reactive power injections at bus . Given the admittance matrix of the microgrid, the active and reactive power injections are related to the voltage profile by the power flow equations [35]

 {Pi(x)=Ei∑j∈VYijEjcos(θi−θj−ϕij)Qi(x)=Ei∑j∈VYijEjsin(θi−θj−ϕij), (1)

where and are the magnitude and the phase angle of the admittance matrix element .

We distinguish the voltage at inverter and load buses in our formulation due to their different characteristics. For inverter buses, there are standard methods to control the voltage magnitude and frequency ([36], [37]). Typically, these methods can track a given inverter voltage reference almost instantaneously. Therefore, an inverter is often modeled as a controlled voltage source behind a reactance [38]. We also adopt such a model and consider can be fully controlled. In contrast to , voltage at load buses is uncertain. The voltage dynamics of the load buses are assumed to satisfy the following condition

###### Assumption 1

, for some

where is a constant determined by load properties and microgrid topology. More explanation about this assumption is provided in Appendix A. Throughout this work, we will focus on constant power or constant impedance loads so that Assumption 1 holds.

###### Remark 1

Under multi-agent or centralized control framework, load voltage is often assumed to be measurable and known during controller design [39, 40]. In this paper, we consider a more general scenario, where load voltage is viewed as an unknown variables with only Assumption 1 being involved in the controller design.

Given nominal active and reactive power injections and , , it is desired that the power injection of the inverters share uncertain loads proportionally to their nominal value:

###### Definition 1

(Proportional Power Sharing) The active and reactive power are proportionally shared among the buses if

 Pj(x)P∗j=Pk(x)P∗k,Qj(x)Q∗j=Qk(x)Q∗k,j,k∈VI. (2)

The power sharing condition (1) imposes a constraint on the system voltage profile. We define this constraint set as

 x∈XS:={x|Eq. (???) % holds}. (3)

In addition to condition (2), the control and operation of a microgrid has to respect its branch angle difference limits and voltage magnitude constraint. The branch angle difference between all connected buses is typically required to be bounded by a given constant . The upper bound is derived based on the maximum current allowable on each distribution line (see [41] and [42]). In addition to branch angle difference, the voltage magnitude also needs to stay inside some secure operation range [35]. Denote as a given desired range of the voltage magnitude of bus . Both branch angle difference and voltage magnitude requirements impose a constraint on the system voltage profile defined below:

 XΘ={x||θi−θj|≤γ,∀{i,j}∈E}, XE={x|E––i≤Ei≤¯Ei∀i∈V},

These two constraints will be referred to as the security constraint of the microgrid

###### Definition 2

(Security Constraints) We say that a microgrid satisfies the security constraints if

 x∈Xc:=XE∩XΘ. (4)

In addition to proportional power sharing and voltage regulation, another important microgrid control objective is known as frequency regulation. Frequency regulation is defined as synchronization without deviations from the nominal value, specifically, , where is a predefined nominal frequency of the microgrid. Note that the power flow equations (2) and the security constraints are invariant with respect to rigid rotation of of all buses. We can select a reference frame rotating at angular frequency while preserving all properties in Eq. (2)-(4). With the rotating reference, the frequency regulation condition is reduced to

 ˙θi=0,∀i∈V. (5)

As discussed in [17], the controller design to meet the requirements specified in Eq. (2)-(5) inevitably requires communication networks. In this paper, we employ a distributed communication structure similar to [12] and [29], where each inverter can communicate with its neighboring inverters to share its local measurements as shown in Fig. 1. Let be a connected simple graph of the communication network, where each edge represents an available communication link between buses and . Let be the set of neighbors of bus , (including bus itself). An inverter has access to the measurements at every inverter bus , including and .

Since each inverter is modeled as a VSI, the microgrid-level coordination control for each inverter reduces to the determination of appropriate voltage frequency and magnitude setpoints. The actual frequency and magnitude can track these setpoints almost instantaneously. The challenge here lies in that the load is uncertain and different load conditions require different voltage profile in order to satisfy constraints (2)-(5). Define , and . Our goal is thus to design a controller for each inverter that can automatically find the desired voltage vector based on local information . Towards this end, we propose to dynamically update as follows

 {˙xi=μi(SNi(x),ENi),∀i∈VIsubj. to x(t)∈Xc,∀t≥0, (6)

where is the control law of inverter to be designed. Note that the above control structure corresponds to directly assigning frequency based on local information, while dynamically updating voltage magnitude through simple integrator dynamics. Such structure is commonly used in the literature of microgrid control, (see [12] and [43]). The constraint is imposed to ensure that the security constraints are always satisfied.

Define , then conditions (2) and (4) hold when . Under Assumption 1, condition (5) holds when for all . Our goal becomes to designing such that forms an equilibrium set of system (6). In addition, we also want to achieve an exponential convergence to for some initial . If such a controller is found, it can steer the microgrid to the desired steady state where conditions (2)-(5) hold. In the rest of this paper, we will first develop the control law such that forms the equilibrium set of the system (6), and then derive conditions to ensure the exponential convergence of .

## 3 A Distributed Microgrid Control Framework

In this section, we propose a distributed control framework to coordinate the inverters in an island AC microgrid to accomplish the control objectives (2)-(5). We first provide sufficient conditions to ensure . A control design framework is then developed.

### 3.1 Existence of Solutions

A minimum requirement for the controller design is the existence of a voltage profile satisfying all the constraints, i.e., . Existing methods in the literature often directly assume this condition holds ([20], [22]). Here, we provide a brief discussion and a set of sufficient conditions to guarantee the non-emptiness of . The existence of the voltage profile satisfying conditions (2) and (4) involves solving nonlinear algebraic power flow equations (2). We revisit a classical result in the following.

###### Lemma 1

[41] Suppose that the following conditions hold

1. The microgrid is connected,

2. The admittance matrix Y is symmetric,

3. for all ,

4. for all ,

5. for all and the strict inequality holds for at least one ,

6. ,

where and are constants determined by microgrid parameters including line impedance and bounds of voltage regulation. Then there exists a solution to Eq. (2) such that .

Lemma 1 is in fact a direct consequence of Theorem 4 in [41]. Readers are referred to [41] for the proof and details of finding and .

###### Remark 2

For simplicity, several conditions in [41] related to loads serviceability are not included in Lemma 1. Since a transition to island mode is enabled only when the DERs can provide sufficient power to loads in the microgrid, the serviceability requirement is satisfied intrinsically for this work.

###### Remark 3

If the nominal active power injection at inverter buses satisfying , Lemma 1 implies . Since we focus on microgrid control problem with given and , we thus assume that and selected in the tertiary control layer are chosen such that . In this way, is nonempty if conditions in Lemma 1 hold. We can then focus on designing controller to steer the microgrid to .

### 3.2 Distributed Controller Design

We start our controller design from a simple property of a connected graph . Let be the Laplacian of . The null space of is span because is connected. Observing that has a close relation with condition (2), we design as a simple linear feedback in terms of in the following form

 (7)

where is the local control gain matrix at bus to be designed. Define , and . Let , and . The dynamical model of the microgrid under the proposed inverter control (7) becomes

 {˙xI(t)=K¯LSI(x(t))subj. to x(t)∈Xc,∀t≥0, (8)
###### Remark 4

The proposed control structure (8) does not depend on the voltage magnitude information that is also available at bus . We will show later that such a control structure is already sufficient to ensure convergence to . In principle, the magnitude information can be used to further improve the control performance, especially for voltage regulation. However, we will not study such extension in this paper.

Define , where , and . The following proposition shows that under some mild conditions, every equilibrium point of system (8) satisfies the control objectives (2)-(5).

###### Proposition 1

If Assumption 1 holds and null(), the following statements are equivalent

1. The microgrid with dynamics (8) is in steady state where .

2. The desired conditions (2)-(5) hold.

Proof. Since null, null(). Null follows directly from null() and null() With null(), we have

 ˙xI=0 such that x∈Xc (9) ⇔ x∈{x|SI(x)∈O}∩Xc⇔x∈Xe,

The equivalence between statements (a) and (b) follows from Eq. (9).

Proposition 1 reduces the microgrid control problem with numerous requirements to the study of exponential convergence to of system (8). We will therefore focus on analyzing system (8).

### 3.3 Analysis of System (8)

In this subsection, we derive the conditions of exponential convergence to where all the desired conditions (2)-(5) follow. The exponential convergence to of system (8) is challenging in general due to the nonlinearity of the underlying system and the uncertainty of the load bus states . Instead of directly analyzing system (8), we apply the chain rule to obtain the dynamics of under the proposed control strategy

 {˙SI(x(t))=JI,x(t)K¯LSI(x(t))+JL,x(t)˙xL(t)x(t)∈Xc,∀t≥0, (10)

where and are the Jacobian matrices of evaluated at with respect to and , respectively. Notice that at every time instant and for any , Eq. (10) describes the dynamics of when the dynamics of is given by Eq. (8). According to Proposition 1, the convergence of to of system (10) implies the convergence of the state trajectory to . The close relation between these two stability properties motivates us to focus on system (10). To simplify notation, we define , , and . System (10) can then be written as a linear time varying (LTV) system

 {˙z(t)=B(t)K¯Lz(t)+w(t)B(t)∈J, (11)

where is considered as a disturbance of system (11). With this notation, system (11) becomes a stand alone dynamic system with state variable subject to unknown disturbance . Note that in system (11), is quadratically bounded by

 ||w||2 =||JL,x˙xL||2≤κ||JL,x||2||˙xI||2 =κ||JL,x||2||K¯Lz||2≤ζ||¯Lz||2 =ζ⋅d(z,O),

where is a constant depending on system parameters as well as control gain . Robust stability of the equilibriums of systems with bounded noise was studied in [44], which is reviewed in the following

###### Definition 3

The set is robustly stable of system (11) with degree if is globally exponentially stable for all such that .

To analyze robust stability for of system (11), we apply a standard change of coordinates. Define a change of coordinate matrix equation

 T=[v1,v2,..,v2n−2,vp∥vp∥2,vq||vq||2], (12)

where the first vectors are arbitrary vectors such that is an orthogonal matrix. Let be the state vector in the new coordinate system. The LTV system (11) becomes

 {˙¯z(t)=T−1B(t)K¯LT¯z(t)+T−1w(t)B(t)∈J. (13)

Since the last two coordinates of the new basis span , the last two column vectors of are zeros and

 T−1B(t)K¯LT=[^A11(t)0^A21(t)0], (14)

where and . Considering that the dynamics of of system (11) is irrelevant to the last two coordinates of the state of system (13), we focus on a reduced order system of (13) with state vector . Define and , we have a reduced order system of (13)

 {˙^z(t)=^A11(t)^z(t)+^w(t)^A11(t)∈A, (15)

where . Similar to in system (11), is quadratically bounded by the state shown in the following

 ||^w||2 ≤||w||2≤ζ||^z||2.

The following lemma shows that exponential convergence to of system (8) follows if the origin of system (15) is robustly stable.

###### Lemma 2

If the origin of system (15) is robustly stable with degree , then there exists an non-empty such that for all , exponentially converges to for system (8).

Proof. Since systems (15) and (13) share the same dynamics in the space , robust stability of the origin of system (15) implies is robustly stable of system (13) with degree . In addition, robust stability of of system (11) (or system (13) ) guarantees the trajectory of is bounded. Define . With the bounded trajectory of , there exists such that for all , , which implies . Therefore, for all initial , converges to with for all time in system (11) if system (15) is robustly stable. Recall that Eq. (11) describes the dynamics of when the dynamics of is given by Eq. (8). We then conclude that for all , exponentially converges to for system (8) if the origin of system (15) is robustly stable.

We now provide a set of sufficient conditions for robust stability of the origin of system (15)

###### Proposition 2

The origin of system (15) is robustly stable with degree if there exist , such that

 [^AT11U+U^A11+ϵζI+ξUUU−ϵI]⪯0, (16)

for all .

Proof. The proof is similar to linear time invariant system case discussed in [44]. Eq. (16) can be derived by quadratic Lyapunov function argument. If there exist a Lyapunov function such that for all , , then the origin of system (15) is exponentially stable. The conditions for , such that are shown in the following

 ˙V(^z)≤−ξV s.t. ||w||2≤ζ||^z||2 ⟺ [^zT^wT][^AT11U+U^A11+ξUUU0][^z^w]≤0 s.t. [^zT^wT][−ζI00I][^z^w]≤0 ⟺ [^zT^wT][^AT11U+U^A11+ϵζI+ξUUU−ϵI][^z^w]≤0.

for all . S-procedure is applied for the last step, which completes the proof.

Note that if is polytopic, the condition (16) can be formulated into bilinear matrix inequalities (BMIs). The condition can then be checked numerically. However, is not polytopic, so we will instead develop a way to find a convex set containing in the next subsection.

With Lemma 2 and Proposition 2, conditions of exponential convergence to of (8) can be obtained:

###### Theorem 1

If Assumption 1, hypotheses in Lemma 1 and Eq. (16) hold, then there exists an non-empty such that for , the microgrid (8) converges exponentially to the set where the control objectives including (2)-(5) are all satisfied.

Proof. The origin of system (15) is robustly stable due to Proposition 2. By Lemma 2, robust stability of system (15) implies the existence of such that for all , the trajectories converge to of system (8). Since Eq. (16) ensures null(), is equivalent to desired control objectives (2)-(5) from Proposition 1 .

The result of Theorem 1 is robust with respect to small variations of system parameters. As long as the perturbations of the admittance matrix are small enough such that , the exponential convergence to for some still follows from Theorem 1. Different from most of the literature, the proposed controller can be applied to mixed ratio distribution lines and general microgrid topology including acyclic and mesh networks. Furthermore, the controller can meet all the main control objectives without the separation of time scale, which distinguishes it from the mainstream droop control methods.

### 3.4 Feedback Gain Design

In this subsection, we propose a constructive way to find a feedback gain satisfying Eq. (16). The difficulty lies in checking the feasibility Eq. (16). As discussed in the last subsection, the robust stability condition in Eq. (16) can not be directly formulated into BMIs because is not polytopic. We will first derive a convex hull containing by analyzing the Jacobian of the power flow equations (2) so that Eq. (16) can be checked by solving several BMIs. Secondly, instead of only checking the feasibility, we formulate the BMIs into an optimization problem to enhance the robustness.

Define and as a polyhedron replacing the constraint in by for all . Let . An approximated convex hull can be found by the following Proposition.

###### Proposition 3

If every entry of the admittance matrix Y satisfies , the upper and lower bounds of every entry of are

 ¯JI,x(i,j)=max{JI,z(i,j),z∈Z}, (17) J––I,x(i,j)=min{JI,z(i,j),z∈Z},

where .

Proof. According to the power flow equation (2), the derivative of the active power injection at bus with respect to different variants are

 ∂Pi(x)∂θi=−Ei∑j∈V∖iYijEjsin(θi−θj−ϕij),∂Pi(x)∂Ei=2EiYiicos(−ϕii)+∑j∈V∖iYijEjcos(θi−θj−ϕij),∂Pi(x)∂θj=EiYijEjsin(θi−θj−ϕij),j≠i,∂Pi(x)∂Ej=EiYijcos(θi−θj−ϕij),j≠i. (18)

Since for all entries in the admittance matrix Y, , every summand of the Jacobian elements in (18) has the maximal and minimal points at if . In addition, every summand corresponds to different set of and in each Jacobian element in (18), the Jacobian elements have the maximal and minimal points at if . The same conclusion for the reactive part is reached by a similar argument.

Given the upper and lower bounds of every entry of , a convex hull containing can be found. Let be a set . Define such that , where . Define such that

 Dϱi(j,k)=¯JI,x(j,k) if ϱi(j,k)=1, Dϱi(j,k)=J––I,x(j,k) if ϱi(j,k)=−1.

The following lemma is a simple consequence of Proposition 3 and provides a convex hull containing .

###### Lemma 3

If every entry of the admittance matrix Y satisfy , the convex hull

 ¯D:=CO{Dϱ1,...,Dϱl}

contains .

The results of Lemma 3 allows us to replace Eq. (16) in Theorem 1 by BMIs. Instead of only finding such that Eq. (16) is feasible, we propose to design by solving the following optimization problem subject to BMI constraints

 maximizeK,U,ϵ,ξ ξ (19) subject to Mi(ϵ,ζ,ξ)⪯0,

where and

 ^A11,i=[I,0]T−1DϱiK¯LT[I,0]T, Mi(⋅)=[^AT11,iU+U^A11,i+ϵζI+ξUUU−ϵI], U=UT≻0,i∈[l].
###### Remark 5

The maximization of is for the purpose of improving the convergence rate. Notice that maximizing is only meaningful if an upper bound of is imposed. More discussions on the upper bound will be included in the next section.

The following corollary is a direct consequence of Eq. (19) and Theorem 1

###### Corollary 1

If Assumption 1, the hypotheses in Lemma 1 hold, and the optimal solution for Eq. (19) is implemented in Eq. (8), then there exists an non-empty such that for , the microgrid (8) converges exponentially to the set where the control objectives including (2)-(5) are all satisfied.

We want to comment that the convex optimization problem (19) subject to BMI constraints is NP-hard to solve in general. However, efficient algorithms [45] and [46] are available if an initial feasible solution can be found. We can first substitute by some simple positive definite matrices as an initial guess and check the feasibility. If is a feasible solution of Eq. (19), algorithms [45] or [46] can be applied to find a local optimal solution. Those algorithms only involve several linear matrix inequalities (LMIs) instead of BMIs, which can be effectively solved by using various existing convex optimization algorithms [47, 48].

###### Remark 6

(Trade-off between Complexity and Robustness) Let . The convex hull does not necessarily contain . However, there exists a compact set such that . Hence, if one solves Eq. (19) by replacing with , Corollary 1 still implies exponential convergence to a subset of for some initial . The benefit of replacing with is reduction of complexity, where while . The difference becomes evident for large and one may prefer to solve the optimization problem (19) by substituting to for microgrids with larger .

As discussed in Remark 6, the number of the BMI constraints in the optimization problem (19) grows exponentially with respect to . Finding for microgrids with large number of bus can become challenging even one replacing by . Here, we observe that matrices in are block diagonal. The number of blocks equals to the number of groups of inverter bus separated by load bus. For example, all matrices of of IEEE 14 bus system shown in Fig. 2 has three diagonal blocks. One of the blocks corresponds to inverter buses, and the values of the entries depend on the states of buses, which are covered with the same color in Fig. 2. Similar argument applies to the other two blocks.

Following the motivating example above, we introduce several notations to define properties of every block in . Let be the set of inverter buses associated with block . The dimension of each block is then given by . Similar arguments to Proposition 3 and Lemma 3 are made to define the convex hull associated with block . The number of vertices of is then . Notice that the large number of BMIs in Eq. (19) is originated from the number of vertices of the convex hull . In the following lemma, we will show that each block in the block diagonal Jacobian matrix can be viewed separately in finding a feasible solution of Eq. (19), resulting in much less number of BMIs involved.

###### Lemma 4

Every that satisfies Eq. (20) is a feasible solution of Eq. (19).

 λ––(Dϱk,iKci+KTciDTϱk,i)≤−d<0 (20) ∀k∈[li],∀i∈[nc],

where and is the number of diagonal block for matrices in .

Proof. From Eq. (20), there exist such that

 λ––(^AT11,iU+U^A11,i)≤−¯d<0,∀i∈[l].

By Schur compliment, if and only if

 (^AT11,iU+U^A11,i+ξU)+ϵ(UU+ζI)⪯0. (21)

We can find such that Eq. (21) holds when and the desired result follows.

Lemma 4 allows us to find by solving Eq. (20) instead of Eq. (19). The computational complexity is greatly reduced because the number of BMIs associated to Eq. (20) increases with respect to instead of . In most microgrid networks, is much smaller than . Solving Eq. (20) instead of Eq. (19) enhances the practicability of the proposed distributed control method, especially for large-scale microgrids.

## 4 Flexible Operation of Microgrids

In this section, we study the plug-and-play feature of DERs under the proposed distributed control framework. The robustness under communication failure is also analyzed to solidify the proposed distributed controller.

### 4.1 Plug-and-Play Feature

The plug-and-play feature of the DERs refers to the property that one DER can be plugged or unplugged to a microgrid without re-engineering the entire control. We consider a general case where part of inverters may be disconnected from the microgrid abruptly due to some severe events. Let be a set of normal operating inverter buses so that is the set of disconnected inverters. The voltage magnitude and phase angle dynamics at buses become unknown and are categorized as load buses. For this reason, we will partition the buses by , for which contains all the normal operating inverters, while consists of all the other buses including load buses or disconnected inverter buses . Consider the case that the communication network remains intact when some inverters are disconnected from the microgrid. The communication graph between the operating inverters is defined as , where . Let be the Laplacian of . The control law of the “fault” microgrid is reduced from Eq. (8) to

 (22)

Notice that due to the assumption on intact communication network, the control law can be autonomously transformed to Eq. (22) in response to the change of microgrid operating conditions. Let and . The microgrid dynamics with the controller (22) is rewritten as follows

 {˙xIf(t)=KIf¯LfSIf(x(t))subj. to x(t)∈Xc,∀t≥0, (23)

where , and . Denoted as the proportional power sharing space of the inverters . If stays connected, the null space of remains equivalent to . The “reduced” microgrid (23) can therefore be analyzed through a similar way discussed in the last section. Define as Jacobians of when and as the number of vertices of . The control objectives (2)-(5) of the reduced microgrid follow if all the hypotheses in Corollary 1 hold except that Eq. (19) is replaced by the following

 maximizeϵf,Kf,U