An energy sharing scheme based on distributed average value estimations for islanded AC microgrids

Electrical Power and Energy Systems 116 (2020) 105587

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An energy sharing scheme based on distributed average value estimations for islanded AC microgrids

T

Khurram Hashmia,b,c, , Muhammad Mansoor Khana,b, Muhammad Umair Shahida,b,d, Arshad Nawaza, Asad Khana,b, Jia June, Houjun Tanga,b ⁎

a

School of Electronics, Information, and Electrical Engineering, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, China Key Laboratory of Control of Power Transmission and Transformation, Ministry of Education, Shanghai Jiao Tong University, 800 Dongchuan RD., Shanghai 200240, China c Department of Electrical Engineering, University of Engineering and Technology, Lahore, Pakistan d Department of Electrical Engineering, Khawaja Fareed University of Engineering & Information Technology, Pakistan e State Grid, Taizhou Power Supply Company, Taizhou, Jiangsu, China b

ARTICLE INFO

ABSTRACT

Keywords: Microgrid control Power electronics systems DC-AC power converters Hierarchical control systems Distributed control systems Distributed average estimation Power system operation

Autonomous, self-governing micro energy networks are a smart energy solution that satisfy the energy needs of isolated communities. Such smart micro networks usually comprise of several distributed semi-controllable power resources, storages and loads. Usually, the power generation sources draw their energy from renewable energy resources. Power electronics converters suitably condition the power to be transmitted and utilized. A sensory, control and communication layer ensures the integration of several network components to achieve stable network operation. Distributed power-sharing methods function at local processing and decision-making nodes across the energy network. The complete control scheme is a hierarchical structure divided into four control levels. Faults and latencies experienced at the communication network layer adversely affect the performance of the control scheme. This paper proposes a new and improved hierarchical, multi-agents-based control strategy for efficient power-sharing, voltage and frequency regulation between nodes during communication link latencies. A novel distributed averaging estimator is proposed to estimate power injected in the system at various nodes and thereby mitigate the effect of relayed information delays. The effectiveness of the proposed strategy is established comprehensively through mathematical modeling, analytical proofs, discretetime small signal stability analysis, and MATLAB case study simulations.

1. Introduction Small scale autonomous distribution networks or Microgrids (MGs) are emerging as a viable solution for widespread integration of distributed renewable energy resources (RES). Distributed generation units (DGUs) using renewable resources have a variable stochastic output [1,2]. Additional supervisory controls are laid out to correct systemic deviations and smoothen integration with the larger power network. Conversely, a “microgrid” can function as a smaller power network independently from the main power grid. In such a situation, power is produced and consumed locally [3]. A microgrid can be composed of several DGUs, energy storage systems (ESS), plug-in hybrid electric vehicles (PHEV), data acquisition and supervisory control devices, localized and centralized controller units [1–4].

Power converters transform power from DGUs for grid integration. Power-sharing between DG nodes can be realized using various powersharing control strategies [5]: Droop control is a set of power-sharing methods based on proportional reduction in voltage and frequency magnitudes to realize active power and reactive power sharing between units. There can be several variations of basic droop methods for different inductive and resistive nature of microgrids [5–7]. A secondary control layer is spread over the primary controls to correct these deviations in frequency and voltage and regulate thee to a nominal range [5,8–14]. A communication dependant control using power and voltage observers for power-sharing between distributed generation units along with voltage restoration and frequency synchronism are presented in [5,15,16]. The drawback of such methods is that they are heavily

⁎ Corresponding author at: School of Electronics, Information, and Electrical Engineering (SEIEE), Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, China. E-mail address: [email protected] (K. Hashmi).

https://doi.org/10.1016/j.ijepes.2019.105587 Received 10 April 2019; Received in revised form 4 September 2019; Accepted 28 September 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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reliant on the transmission of data. These Schemes are modelled as a hierarchical control system implemented either as a centralized controller or through decentralized controllers at each power conversion unit. Centralized control strategies typically require two-way communication with a microgrid central controller (MGCC). Although centralized control strategies can be easier to implement, the MGCC stands out as a single point of failure (SPOF) in the system. Furthermore, if not aided by local controllers, the central controller will face a high processing burden. Conversely, decentralized methods are far more flexible and agree well with innovative new control schemes based on artificial intelligence and multi-agent systems consensus [18–22]. Multi-agentbased control methods enable every DGU node to participate in the decision-making process. The microgrid operation and control problem can be formulated such that each power converter can behave as an agent within a multiagent systems framework. Power-sharing, voltage and frequency restoration problems are formulated as tracker synchronization problems where all nodes strive to reach a consensus on corrective measures [23,24]. Power network regulation can be realized through a hierarchical multi-layered control structure. The lowest “zero level” controls deal with voltage and current regulation at every inverter. The “primary” layer controls power-sharing between nodes. The secondary layer overlaps the zero and primary levels and is dedicated to restoring frequency and voltage deviations. The tertiary layer facilitates the powersharing with main grid through the voltage and frequency regulation at the point of common coupling (PCC) [25–27]. The secondary control layer is designed on a multiagent consensus principle that sees frequency and voltage restoration problem to a tracker synchronization distributed problem. The power-sharing problem can be modelled as a distributed consensus control goal [27]. Reactive power-sharing is addressed as a primary goal through a distributed control approach that works towards a trade-off with voltage restoration [28,29]. System stability is analyzed through small-signal models using distributed control algorithms for power-sharing, voltage and frequency restoration. Consensus-based distributed voltage control algorithm to regulate voltage, and sharing reactive power in meshed micro-grids to replace traditional V-Q droop method are presented in [30]. A distributed cooperative control strategy for microgrids is presented that proposes alternate to centralized secondary control and the primary droop control of each inverter in [15]. Voltage, reactive power, and active power regulators are designed to regulate these parameters. A droop based method for cooperative control of microgrids with a variable overlaying communication network is investigated in [31]. Consensus-based controllers for active and reactive power, voltage and frequency regulation are used to implement this control scheme. Alternatively, an iterative event-triggered control scheme for multi-energy resource DC microgrids with limited communication bandwidth for the control layer is presented in [32]. The Lyapunov technique is used to derive event-triggered conditions to verify efficacy of the scheme. A neighbor based distributed cooperative control scheme to regulate voltage in DC microgrids is presented in [33]. The consensus-based methods critically rely on the health of communication links used by the control network. Faster transmission of measurements and control signals translate into earlier convergence and improved system regulation, whereas, intermittent and latent communication links cause deterioration in control performance [34,35]. To simplify problem formulation and control design most researchers assume a fault-free communication network that has no broken, disrupted or latent communication links as in [17,24,25,28,30]. The communication networks represented in such system models are, therefore, time in-varying. Some studies, such as [15,27,31–33,36] consider scenarios with faulty communication links and present some methods to mitigate their effect. In practical systems, communication intermittencies in wired and wireless networks are inevitable and common. The control methods

based solely on multiagent consensus and communication of values are severely affected due to delays in communication signals. Therefore, to address the problems created by the delayed transmission of key parameter estimates and control signals, an innovative and rational strategy is required. In this work, the authors propose a new and improved hierarchical control scheme to deal with the problem of latent communication links in the control of islanded AC microgrid networks. The control strategy presented combines the use of novel distributed average estimators at every intelligent power conversion node in the system. These estimators arrive at average values of active and reactive power at every node, which serves as nominal references for the power controllers at each power converter. This method adds sufficient resilience to communication latencies and errors in the system. The proposed strategy is compared with conventional consensus-based power-sharing controls to establish its relative merit. Comprehensive mathematical analysis, analytical comparisons and numerical simulation of the proposed method against conventional consensus methods is provided for verification and closure. The major contributions of this work, therefore are: (1) Novel finite impulse response averaging estimators are developed for estimating the average active and reactive powers injected at each node. These estimates are sufficiently resilient to communication link delays. (2) Improved power-sharing controls are implemented that perform satisfactorily under communication link delays. (3) A hierarchical distributed control structure is created that functions without a centralized monitoring unit. (4) A composite small-signal model of the complete MG system under the proposed control scheme is derived and analyzed. (5) Detailed analysis and comparison of the proposed methodology with conventional consensus-based power-sharing methods is presented under the effect of communication latencies. The remainder of this paper is organized as follows. Section 2 provides the problem description with details of the system used in this study. Section 3 mathematically outlines the proposed distributed averaging method for power-sharing and provides detailed analytical comparisons with conventional consensus-based method. Voltage and frequency restoration controls techniques are explained in Section 4. Section 5 illustrates the derivation of the small-signal model for this microgrid system with the proposed controls. Section 6 gives details on the discretization of the derived system model. Section 7 elaborates stability analysis of the system under the proposed control method in comparison with the conventional consensus-based method. System performance evaluation with simulation studies are discussed in Section 8, and finally, Section 9 concludes this paper. 2. Problem description This section describes the microgrid system considered in this study. The problem formulation is outlined and the framework for this study is introduced. The smart micro-network is composed of two overlapped, dynamically interacting systems: (i) the power distribution network and (ii) The communication and control infrastructure network. Therefore, every power injecting DGU is considered as a participant node or agent in a larger multiagent system. To adequately describe the problem and propose the control strategy, we systematically go about describing component theories, such as the basic notions of graph theory, power systems steady-state equations along with the necessary mathematical representations. A description of communication latencies is presented. 2.1. Power distribution network Fig. 1 shows a radial distribution network used in this study with 2

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Fig. 1. Microgrid Power Distribution and Control Network. Table 1 System Parameters.

Table 2 Connected Loads.

Parameters

Values

Parameters

Values

Lf Rf Cf Lc Rc Rline Lline fnom Vnom

1.35 mH 0.1 Ω 25 µF 1.35 mH 0.05 Ω 0.1 Ω 0.5mH 50 Hz 415 VL-L

mp nq Kpf Kif KpV KiV F ωc

4.5e-6 1e-6 0.4 0.5 0.5 0.3 1 50 Hz

Bus. No.

Bus wise Connected Loads

1. 2. 3. 4. 5. 6. 7.

P (p. u.)

Q (p. u.)

0 0.3 0.25 0.25 0.25 0.25 0

0 0.3 0.25 0.25 0.25 0.25 0

The Eqs. (1) and (2) represent active and reactive powers injected at each node, respectively. The simplified steady-state model of the power network is given by Eq. (3). The bus admittance matrix, YbusMG of distribution network is provided in the Appendix A.3.

three-phase, three-wire configurations. Power converters are connected to Buses 1 through 6 using LC filters and coupling inductors. Variable RL loads are connected to buses 2 through 6, whereas, bus-1 is not loaded directly. Bus 7 provides a point of common coupling (PCC) with the main power grid. This network can be operated autonomously in an “islanded” mode. Table 1 outlines rated system parameters and Table 2 provides system loads.

Pi =

3

N n=1

|Yin Vi Vn|.cos(

in

+

n

i)

(1)

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Qi =

N n=1

|Yin Vi Vn|.sin(

in

+

n

i)

EG VG × VG . The adjacency matrix associated is given by AG = [aij] RN × N . In a smart microgrid, the DGUs can be modeled as the nodes of a communication digraph whereas the arcs model communication links [37]. Fig. 2a elaborates the interaction between primary and secondary control layers at a single node. voltage and frequency corrections (ωni, Vni) are generated through observer consensus based on global voltage and frequency references (vref , ref ) as well as neighborhood estimates (voj , oj ) . The third and fourth components estimate injected power average values (P , Q ) , based on neighbor estimates ( Pavg . N , Qavg . N ) , as will be elaborated mathematically in Section 3. A stable communication network is considered initially that can be described as time-invariant as represented in Fig. 2b. Transmission channel noise may be neglected for simplifying calculations. The resulting digraph is time-invariant, i.e. Ag is a constant. An arc from node j to node i is denoted by (vj, vi ), where node j receives information from node i. aij is the weight of the arc connecting vi to vj. aij > 0 if (vj, vi ) Eg , otherwise aij = 0 . Node i is called a neighbor of node j if the arc (vj, vi ) Eg . Set of nodes neighboring the ith DGU vi are given by Ni={vj Vg : (vi , vj ) Eg }. The Laplacian Matrix Lg = (lij )N × N is defined N as lij = aij , i j and lii = j 1 aij for i = 1, .,N . Such that L1N = 0 T N R . The in-degree matrix can be defined as with 1N = (1, , 1) DGin = diag {diin} , where, diin = j Ni (aji ) and out-degree matrix as DGout = diag {diout } , where diout = i Ni (aji) . A multiagent observer consensus algorithm implemented across the networked control system causes system states x to converge over a span of time, that can be represented as Eqs. (4) and (5).

(2)

[YbusMG]· [V1 V2 V3 V4 V5 V6 V7 ]T = [ Is1 Is 2 Is3 Is 4 Is5 Is6 Is7 ]T

(3)

th

where Yin represents the total admittance between the i and nth buses; Vi represents the magnitude of voltage measured at ith inverter terminals. The magnitude of voltage at the nth bus is represented as Vn; the admittance angle between bus ith and nth bus is represented as in , th n is the angle of voltage at n bus whereas i is the angle of voltage at th the i bus. 2.2. Hierarchical controls and information sharing network The control scheme for the microgrid is described as a hierarchical control structure composed of three broad layers, graphically represented as in Fig. 1: the inner or zero level control loops regulate voltage and current control for each power converter, generating gate pulses to operate individual converters. The primary layer regulates power-sharing using droop control technique using droop gains (mP*, nQ*). The secondary/ tertiary layer is composed of power, voltage and frequency observers that estimate these parameters based on information from neighborhood nodes ( k N oi , k N voi, k N Pdist . g , k N Qdist . g ) , thereby providing regular corrective updates for power injection, voltage and frequency values(P , Q , ni , Vni ) to maintain these within agreeable range on a system-wide scale. 2.2.1. Graph theory preliminaries The communication network layer serves as the media for measured and estimated values and control signals to travel through the control system based on connectivity structure as shown in Fig. 2b. This network can be modeled as a digraph expressible as Gcom = (VG, EG, AG ) composed of a non-empty, finite set of M nodes given as VG = {v1, v2, v3, , vM } . The arcs that link these nodes are given by

x =

C (DG

AG ) x =

CLG x

(4)

where

LG = DG

AG

(5)

Eq. (5) represents the Laplacian matrix LG that is calculated for the

Fig. 2. Distributed control scheme: (a) Secondary controls at every node (b) Full bi-directional connected ring communication network (c) Ring communication network with isolated dual link latencies.

4

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communication network based on adjacency matrix AG and degree matrix DG , x represents the system states and x represents their evolved values as per the consensus algorithm, the factor C is a convergence factor whose value can be modeled based on network parameters [38]. The matrices AG, DG, and LG for the communication topology used are given in the Appendix A.2.

x avg . i (k + 1) =

(k ).

m k )

P¯ )

mPi (Pi

V di = Vdi +

nQi . (Qi

(6)

Vo =

z 1. x m1 (z )(1 z 1. xm2 (z )(1 z 1. xm3 (z )(1

(7)

z 1. xmN (z )(1

x mi = [Pmi, Qmi]

(12)

z1

N)

(13)

1

+ xTdis. g (z ). z

z1 z1 z1

N)

+ xTdis2 (z ). z 1 = xTdis1 (z ) + xTdis1 (z ). z 1 = xTdis2 (z ) N) + x 1 = x Tdis 2 (z ). z Tdis3 (z ) . 1 N 1 z ) + xTdisN 1 (z ). z = xTdisN (z ) N)

(14)

The equations can be compactly written in matrix form as Eq. (15).

xTdis1 (z ) xTdis2 (z ) xTdis3 (z ) . xTdisN (z )

=

0 0 0 0 z 1

0 0 0 z 1 0 0 z 1 0 0 z 1 0 0 . . 0 0 0 0 0 0 0

xTdis .1 xTdis .2 xTdis .3 .

xTdis (N

x m1 x m2 xm3 . z 1. (1 . xmN

+ 1)

z1

N)

(15) For convenience, we may assign the matrices as:

[xTdis (z )]N × 1 =

=

0 0 0 z 1 0 0 z 1 0 0 z 1 0 0 , [xm ]N × 1 . 0 0 0 0 0 0 0

0 0 0 0 z 1

xTdis1 (z ) xTdis2 (z ) xTdis3 (z ) , [Z ]N × N = . xTdisN (z ) x m1 x m2 x m3 . . xmN

(16)

Now, therefore the equations may be re-written as:

(8)

[xTdis (z )] = [Z ]. [xTdis (z )] + [xm]. z 1. (1

z1

N)

(17)

z1

N)

(18)

we may further derive:

s

+

1)

N

Considering a system of N nodes each having a distributed power averaging estimator as shown, we may write a set of N equations in Zdomain as Eq. (14).

Q¯ )

[xTdis (z )] Vqi2

(11)

(N + 1) 1)

xm . i + x dist . g (k

x dist . i (z ) = z 1. xmi (z )(1

Vqi = 0 Vdi2

xdist . i (k + 1)

Let, the locally calculated values of active and reactive power at node i, be represented by xmi, where x mi = [Pmi, Qmi]. The estimate of the summation of all injected powers received from neighboring nodes is given by xdist.g, where x dist . g = [Pdist . i, Qdist . i]. The estimate of summations of powers calculated at node i can be given by Eq. (13)

A novel power-sharing control is proposed to overcome the effect of communication link latencies. This method is based on a guided “droop” principle that draws inspiration from the theory of finite impulse response filters (FIR) [41]. Distributed average power observers/ estimators are programmed at every node as shown in Fig. 2a and Fig. 3a and b. Collectively, all the system nodes arrive at average injected power estimates based on estimates received from neighboring nodes and local measurements. The average power is used as a reference for local droop controls according to the Eqs. (8) and (9), where the frequency and voltage are varied proportionally to achieve sharing of active and reactive power. Eqs. (8)–(12) mathematically represent the proposed power controller. i

1)

N

3.1. Elaboration of distributed estimation of measured power.

3. Distributed average power estimation scheme

=

(N + 1)) + xdist . g (k

where N is the number of nodes, i is the present node. xdist is the averaged power exchanged between nodes: xdist.g is the average received from neighbor node, and xdist.i is the power average sent out by node i. (P , Q ) represent averaged estimates computed at the node.

where km represents the communication delay occurring at time instant k and (k ) is a stochastic binary variable with Prob { (k ) = 1} = pk where pk can take only discrete values [39]. The system convergence is more sensitive towards overall maximum delay encountered in the communication links than to finite, bounded stochastic intermittencies. An analysis considering the stochastic nature of intermittencies becomes more involved due to the consideration of variable time systems. For the study of system convergence maximum, discrete-time delays are sufficient as well as convenient for the study of a large system model. Therefore, In this work, we approximate the communication delay km with a uniform unit time delay represented by td [34,38,40]. This delay is varied incrementally in discrete equal steps to emulate an increasing delay for different scenarios.

i

x m . i (k

xavg . i (k + 1) =

where matrices A, B and C represent a simplified form of the state-space matrices of a networked controlled system in discrete time, x p represents the vector of system states considered in this modelling, up are the inputs and yp are the outputs derived from the system. Assuming that a networked communication system is in place to relay the measured outputs, for a general case we may assume that all parameter measurements after passing through the communication network exhibit a randomly varying communication delay given by Eq. (7)

yc (k ) = yp (k

1)

xm . i = xm . i (k 1) xm . i (k xdist . i = xm . i + x dist . g (k

2.2.2. Information network latencies We consider an information network where communication latencies directly affect three nodes, i.e., DGU 1,6 and 5, as shown in Fig. 2c. The dotted lines here represent communication links having a latency whereas DGUs are labeled 1 to 6 accordingly. The DGU 6 is dually affected by communication health decay since all the links connecting it suffer from communication latencies. To elaborate the modeling of communication latencies, a networked control system can be described in discrete state-space form as in (6):

xp (k + 1) = A. x p + B. up yp = C . x p

x m . i (k

(9)

([I ]

(10)

[Z ]. [xTdis (z )] = [xm]. z 1. (1

[Z ]). [xTdis (z )] = [xm]. z

[xTdis (z )] = ([I ] 5

1.

(1

[Z ]) 1. [xm]. z 1. (1

z1 N )

(19)

z1

(20)

N)

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Fig. 3. (a) Power controller (b) Distributed power averaging estimator (c) Voltage control loop (d) Current control loop.

By simplifying (20), we may write the distributed power estimation as:

xi = xi +

N

xTdis . i (z ) =

(xmi)

x¯i )=x i +

N

x¯i (z ) = x avg =

(22)

N

where x i = [Pi , Qi ] gives average estimates for active and reactive power values at the ith node, and N is the total number of nodes in the system. From the above elaboration, it can be concluded that the proposed method estimates the average of measured power at every node in the system in a distributed way.

sX¯ i = sXi + sAG . e

0

x¯i ( )). d

j Ni

0

aij x¯j (

td) x¯i ( )). d .

j Ni

td s .

(

DGin . x¯i .

(26)

X¯j

DGin X¯ i .

(27)

td s

DGin

) 1X = Hobs X .

(28)

N × N and H where IN obs are the identity matrix and observer transfer function, respectively. Eq. (28) expresses the global dynamics of these power estimators. 1, the Eq. 0 , the term etd s When the time delays are small i.e. td (28) can be re-written as,

X¯ = s (sIN + AG

DGin) 1X .

(29)

And can be further simplified to

(23)

X¯ = s (sIN + L) 1X .

Considering a system with communication delays td and weight factors in the adjacency matrix aij . we can re-write the equation as. t

td )

X¯ = s sIN + AG . e

For the sake of comparison, we alternatively consider an observerbased consensus approach for the primary power-sharing controller [15,42]. This estimates the average power reference x i = [Pi , Qi ] through observation of previous and current neighborhood measurements. Such an observer may be described as in Eq. (23).

aij (x¯j ( )

(25)

where X and X are the Laplace transforms of x and x respectively. With some mathematical manipulations, we can write

3.2. Conventional consensus-based method

t

td ) diin x i .

where (26) gives the dynamics of the decentralized consensus-based power observers; x i = [x1, x2 , .,xN ]T represents a vector of power measurements at all nodes, x i = [x1, x2 , .,xN ]T denotes power estimation vector obtained thorough consensus averaging. Using the Laplace transform to convert these equations into the frequency domain, we get

(xTdis . i )

i=0

aij x¯j (t j Ni

x¯i = x i + AG . x¯j (t

Following these, the average values of power estimated at any node in the system may be given by:

x i (t ) = x i (t ) +

td )

On further simplifying, we get

(21)

i=0

x i (t ) = x i (t ) +

aij (x¯j (t j Ni

(30)

where LG = DG AG is the Laplacian matrix. If the matrix LG is balanced, the elements of x converge to a consensus value which represents a true average of power estimates. In the next section, it will be shown that the convergence behavior is highly dependent on the connectivity of the system.

(24)

Taking time derivatives to obtain system dynamics, we get 6

Electrical Power and Energy Systems 116 (2020) 105587

K. Hashmi, et al.

Fig. 4. A comparison (a) Proposed FIR based estimators (b) Consensus-based observers.

3.3. Comparison of system convergence under consensus and distributed averaging

time for such a control can be described as N . td . As the communication 0 . Unlike consensus-based 0 it is intuitive that N . td delays td controls, the proposed system will converge for a finite number of nodes in negligibly small time. It can be concluded that the proposed FIR based technique is more promising in this scenario.

Consider a simplified consensus-based control as described in Section 3.2 and shown in Fig. 4b. For a fully connected networked 0 , all the system, if communication delays tend towards zero i.e. td system poles lie on the real axis. For a sparsely connected communication network, with no communication delays, a suitable value of AG = [aij] RN × N such that matrix LG is balanced, leads the system towards convergence. However, when communication delays of a span td are encountered in one or more of the links, the term AG . e td s in s (sIN + AG . e td s DGin) 1 causes system zeroes to move into the left half-plane therefore resulting in a non-minimum phase system. For any such system, there will exist values of AG such that the system fails to converge, and therefore [aij ] RN × N may no longer have arbitrary values. For greater values of aij, divergence can be observed even in fully connected systems. Similarly, there exist values of time delay td , that the system fails to converge and visible disturbances appear [37,43]. We may, therefore, conclude that using conventional consensus-based controls with finite communication delays, the overall system convergence will be slower than the span of time delays encountered in the communication links. If the distributed grid system is considered a plant in the model, even smaller values of aij will have to be chosen making the system convergence even slower. Alternately, consider the proposed FIR estimator as shown in Fig. 4a. The finite impulse response filters (FIR) have no analogue transcendental functions. These are therefore derived directly in the discrete-time domain. The FIR constructs the output in relation to a finite input history. The “length” of the filter is composed of the number of delays being considered where each one of the nodes represents a unit time delay [41]. For an FIR of an order N, the convergence of the system to step input can be obtained in N steps. Hence, the convergence

4. Secondary layer controls: voltage and frequency regulation Voltage and frequency regulation are handled by multiagent consensus-based secondary controls [37]. Fig. 2a reveals the distributed frequency and voltage restoration control scheme. The method for frequency regulation given by (31). i (k

k

+ 1) = kpf e i (k ) + kif

e i (k )

i = ko

e i (k + 1) =

j Ni

aij (

oi (k )

oj (k ))+hi ( oi (k )

ref

(k ))

(31)

where nominal reference frequency is given by ref , the measured system frequency is given as oj that is measured at all nodes in the neighborhood of the ith node. The proportional and integral gains of secondary frequency restoration are kpf and kif that are shown in Fig. 2a. The correction term applied to frequency reference of the ith inverter is represented by i . The pinning gain is given as hi whose value is zero for the primary node. The voltage regulation method is described in (32):

Vi (k + 1) = kpv e vi (k ) + kiv

k

e vi (k )

i = ko

e vi (k + 1) =

j Ni

aij (voi (k )

voj (k ))+hi (voi (k )

vref (k ))

(32)

where vnom shows nominal reference voltages for the system in p.u., voj is the system voltage measured at all converter nodes in the 7

Electrical Power and Energy Systems 116 (2020) 105587

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communication neighborhood of the node i being considered. The proportional and integral gains for secondary voltage restoration are given as kpv and kiv as shown in Fig. 2a. The voltage correction terms added to voltage reference of the ith inverter node is given by Vi . The pinning gain is given as gi whose value is zero for the primary node.

to voltage controls. The reference for voqi* is set to zero [17]. The variation in injected power can be represented as the difference between measured and average estimated values at any instant,

[ Pi] = [P¯i] [Pi] [ Qi ] = [Q¯ i] [Qi]

5. Composite model of a power converter based MG

where Pi , Qi are estimates of average active and reactive power flowing at node i and Pi, Qi are values of powers calculated at the node i.

To analyze the proposed distributed control strategy, small-signal analysis of the complete MG system is undertaken. A state-space model of the MG system is described in terms of matrices A, B, C and D [38,39,44]. Large signal equations are perturbed to get a small signal model of the system components and then substituted in place to obtain a model for the entire MG system. This section elaborates each one of the model components used in the development and analysis of the control scheme.

5.3. Mathematical analysis of secondary controls with time delay Following from the secondary control law given in Section 4, secondary voltage and frequency regulation can be modelled using a combined vector of the correction variable [ ] = [ i , vi], where [ i , vi] represent variations in frequency and voltage as:

=

5.1. Inner converter controls

= [0]. [

dq]

+ BV 1. [ vodq] + BV 2. [ ildq

[ ildq] = CV . [

dq]

+ DV 1 vodq + DV 2 . [ ildq

dq]

vodq vodq

A B

=[

d

iodq ]T iodq ]T

(33)

dq]

= [0]. [

[ vidq] = CC [

dq] dq]

=[

d

ki

i (t )

t o

j (t ))

( i( )

j(

)). d

A (t )]

=

[kp]N × N [Dg ][

[

B (t )]

=

[ki]N × N [Dg ][

+ BC1. [

ldq]

+ BC 2. [ ildq

+ DC1 [ ildq] + DC 2 . [ ildq

vodq vodq

[

. (38)

i (t )] i (t )]

+ [kp]N × N [Ag ][ + [ki]N × N [Ag ][

j (t j (t

td)] td)]

.

(39)

[

iodq ] T

iodq ]

]=[

A

]+[

B

(40)

],

]=[ ]+[

(41)

],

The perturbations in frequency in voltage can be given as

T

(42)

] = [ vdqi, wi],

The variations in line and load currents as well as inverter parameters with these perturbations are now written as

(34)

[ ilineDQ] = ANET [ ilineDQ] + B1NET [ vdqi] + B1NET [ wi]

T q]

[ iloadDQ ] = ALOAD [ ilineDQ] + B1LOAD [ vdqi] + B2LOAD [ wi ]

P Q

(43)

The variation in inverter parameters may be updated as

wi + wcom Cinv . wi = . [ xinv . i] ioDQi Cinv . Ci

Small signal equations for the active and reactive power sharing control as shown in Fig. 3a can be written in mathematical form as Eq. (35)

P + BP Q

.

[ xinvi] = Ainvi [ x invi ] + Binvi [ vdqi] + Biwcom [ wi + wcom]

5.2. Power sharing control

w CPw vodq = CPv .

=

kp (

The perturbations are given by

where auxiliary state variables for the PI controllers used are d and q as seen in Fig. 3a. The matrices BV1, BV2, CV1, DV1, DV2, BC1, BC2, CC, DC1, DC2 are given in the Appendix A.5.

P = AP . Q

=

[

[

where dq

(37)

B.

where the correction factors are given by

T q]

where the auxiliary state variables for the PI controllers are given by di and qi . System measurements voqi, void, iodi and ioqi are as seen in Fig. 3c and d. The small signal model for current control loops at each node as shown in Fig. 3b are given as Eq. (34).

[

+

Assuming an information transmission delay of time td is encountered, the combined small signal model for frequency regulation can be modelled as per [38] and represented as

where dq

A

where

Voltage and current controls are formulated in the d-q-0 frame that form local control loops for the power converters as shown in Fig. 3c and d. Large signal dynamical equations are perturbed to get small signal equations for voltage control loop represtend as Eq. (33), [45,46].

[

(36)

. (44)

5.4. Grid-Side filter of power converter The small-signal model for the output filter containing an LC filter and a coupling inductor L can be given by (45).

ildq vodq + BPwcom [ wcom ] iodq

ildq vodq = ALCL iodq

ildq vodq + BLCL1 [ vidq] + BLCL2 [ vbdq] + BLCL3 [ iodq

]

(35)

(45)

The controller matrices Ap, Bp, Bpwcom, Cpw and Cpv contain details of the power control variables and are provided in Appendix A.6. The power controller gives local operating frequency for the DGU (ωi) and reference voltage (components: vodi* and voqi*) that give reference points

The input and output parameters, as seen in Fig. 2a may be transformed to a common rotating d-q-0 reference frame using a transformation matrix T , rotating with angular frequency com as described in Appendix A.4.

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5.5. The ith inverter model

Appendix A.1.

The components elaborated in previous sections can be combined to arrive at a small-signal model of i th distributed generation unit. This model can be represented in the form of Eqs. (46) and (47).

6. Discretization of the system model Consider continuous-time state space and output equations of the above-described systems

[ xinvi] = Ainvi [ x invi ] + Binvi [ v dqi] + Biwcom [ w i + wcom]

x (t ) = Ac . sys . x (t ) + Bc . sys . u (t )

w i + wcom Cinv . wi = . [ x inv . i] ioDQi Cinv . Ci

y (t ) = Cc . sys. x (t ) + Dc . sys . u (t )

(46)

i

Pi

Qi

dqi

ildqi

dqi

vodqi

iodqi ]T .

(52)

where Acsys, Bcsys, Ccsys, and Dcsys represent the continuous-time representation of the MG systems. The discrete-time state-space representation of these equations takes the form:

where the state vector is,

[ x inv ] = [

,

(47)

The matrices Ainvi, Binvi, Biwcom, Cinwi, Cinvci are provided in the Appendix A.1.

x ((k + 1) T ) = Gd (T ) x (kT ) + Hd (T ) u (kT ) , y (kT ) = Cd x (kT ) + Dd u (kT )

5.6. Combined model for MG inverters

where the matrices Gd, Hd,Cd,Dd are discrete-time counterparts for Acsys, Bcsys, Ccsys and Dcsys matrices in continuous-time that can be obtained from (54), (55), and (56).

A combined model for N power converters connected to the microgrid network is presented as in Eq. (48)

[ xinv ] = Ainv . [ x inv ] + Binv . [ vbDQ ] [ ioDQ] = Cinvc . [ x inv ]

H (T ) =

where the state vector is,

[x inv ] = [ xinv1

xinv 2 . . .

vbDQ2 . .

x invN ]

vbDQN ] .

[ iloadDQ ] = ALOAD [ ilineDQ] + B1LOAD [ v dqi] + B2LOAD [ w i ]

(49)

where

ilineDQ = [ ilineDQ1, ilineDQ2, ... ilineDQN ]T iloadDQ = [ iloadDQ1, iloadDQ2 , ... iloadDQp]T vbDQ = [ vbDQ1, vbDQ2, ... vbDQN ]T = com

.

The matrices ANET, B1NET, ALOAD, B1LOAD, and B2LOAD are system matrices described in the Appendix A.1. 5.8. Composite model of micro grid The priory described component models can be combined to give a composite small signal model for the complete microgrid system as event-triggered by Eqs. (50) and (51). The system used in this study has m = 7 nodes, s = 6 DGUs, p = 5 loads, n = 6 lines. The entire model is composed of complex detailed sub-components; therefore, MATLAB Simulink and linear analysis tools are used to evaluate this system.

[vbDQ ] = RN [Minv [ ioDQ] + MLoad [ iloadDQ] + Mnet [ ilineDQ]]

i loadDQ

x inv i lineDQ i loadDQ

)B

c . sys

(55) (56)

The discrete composite model of the MG system is given by Eqs. (53)–(56). This is used to analyze stability of the system under varying communication latencies and control gains. This model is used to obtain limits for the test MG network under the proposed and conventional control schemes. Several key analyses are performed to this end. The power-sharing gains of all power controllers (mP, nQ) are perturbed to arrive at operational limits for the system where the poles begin to move outside the unit circle as observed in Fig. 5. Communication time delays td for selected link pairs linking one system node (a16, a61) and (a56, a65) are varied incrementally to observe the effect produced on system poles and zeros. Fig. 5a and b demonstrate the effect of variation in mP and nQ on system poles under the proposed distributed power averaging method. to observe the variation in poles due to variation in mP, a fixed nominal value of nQ is selected and maintained whilst mP is varied to trace the trajectory of system poles. Fig. 5a presents the results of this variation in mP. Similarly, Fig. 5b elaborates the effect of incremental variation in nQ keeping a fixed nominal value of mP. For comparison with the range of system gains under consensus-based control, Fig. 5c and d present the effect of variation in mP and nQ under conventional consensusbased power estimation methods obtained with the same procedure as described above. For all the cases, the control values (mP, nQ), for which the system poles appear to be in the vicinity of the unit origin are considered to be maximum allowable limits with regards to system stability. Therefore, system stability, sensitivity towards control gain and behavior is predicted using these pole-zero evolutions. The maximum and minimum gains are summarized in Table 3. In comparison with a conventional consensus-based power-sharing approach, the proposed methodology gives more stability to the MG system under varying primary control

[ ilineDQ] = ANET [ ilineDQ] + B1NET [ v dqi] + B1NET [ w i]

= A MG

e Ac .sys d

7. Stability analysis

A model for the distribution network and system loads is derived through Kirchhoff voltage and current laws and can be expressed in terms of line currents and node voltages as in Eq. (49).

x inv

T 0

The matrices Gd and Hd depend on sampling period T. If the sampling period T is fixed, the matrices are constant. Matrices C and D are constant matrices independent of the sampling period T [47]. For delay analyses, the time delays are assumed as greater than the sampling period and being integral multiples of the sampling period.

T

5.7. Network and load model

i lineDQ

(

Cd = Cc . sys Dd = Dc . sys

T,

And,

[ vbDQ ] = [ vbDQ1

(54)

G (T ) = e Ac.sys T

(48)

(53)

(50)

(51)

The system matrices AMG, RN, Minv, Mload, and Mnet are provided in the

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Fig. 5. Pole zero plots:(a) Effect of mP variation with distributed power averaging estimators (b) Effect of nQ variation with distributed power averaging estimators (c) Effect of mP variation with consensus-based observer (d) Effect of nQ variation with consensus-based observer.

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varying time delays with the proposed control scheme. Alternatively, Fig. 6c and d elaborate the behaviour of the proposed MG system under a conventional consensus-based power-sharing approach for incrementally varying time delays. It is observed that for consensus-based controls, there exist values of time delays at td > 1s that cause system poles to move outside the unit circle in stability plots, thereby making the system unstable and divergent. Whereas, the distributed value estimation through FIR filters proposed in this paper, is resilient to such divergence resulting from communication delays. For any value of time delay, the proposed FIR estimator and system poles remain within the unit circle, thereby demonstrating system stability. This verifies the analytical comparison made in Section 3.3 between conventional consensus-based controls and the proposed method. It can be inferred that in comparison with a consensus-based approach, the proposed distributed averaging-based method exhibits system stability in presence of communication latencies. Whereas, after a certain maximum value of time delay, consensus-based controls fail to converge the system to nominal values. It can be further drawn from the analysis made in Sections 3.2, 3.3 and the above stability analyses that consensus-based controls perform slower in a sparsely connected network compared to the proposed FIR based technique. Unidirectional propagation of information through a ring network without communication delays represents a “daisy chain” like structure, where the information is processed at one node first before being sent to the next node and so on. The roots of the characteristic equation of consensus-based controllers greatly depend on the adjacency matrix AG = [aij] RN × N [43]. There is, therefore an upper

Table 3 Variation range for controller gains and system parameters. Sr. No.

Control Parameters

1.

Power Controller mp nq

Min

1 × 10

10

1 × 10

7

Max 1 × 10

5

4 × 10

3

2.

Frequency Regulation kpf kif

0.4 0.1

2.5 0.5

3.

Voltage Regulation kpV kiV

0.5 0.1

3.5 0.5

4.

Time Delay τdelay

0

5s

gains. Among the two active and reactive power sharing gains, the system is more sensitive towards variations in reactive power gain nQ than mP. The reactive power sharing controls, therefore, operate over a narrower margin. Fig. 6a–d demonstrate the behaviour of this MG system under the influence of incrementally varying time delays. Starting with no time delay (td = 0) we move in incremental steps towards a maximal value of time delay (td = 5 s) and record the movement of system poles. Fig. 6a and b demonstrate the pole and zero traces resulting from the behaviour of this MG system under the influence of incrementally

Fig. 6. Comparison of the effect of time delay on system stability: (a) Proposed distributed averaging method (b) Distributed averaging method (magnified) (c) Consensus based power estimation method (d) Consensus based power estimation method (magnified).

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Fig. 7. Active power sharing results: (a) Power sharing with proposed control no link latencies (b) Power sharing with consensus-based control no link latencies (c) Power sharing with proposed control with link latencies (d) Power sharing consensus based with link latencies.

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Fig. 8. Comparison of Secondary Control performance: (a) Frequency restoration with proposed scheme (b) Voltage restoration with proposed scheme (c) Frequency restoration with observer-based control (d) Voltage restoration with observer-based control.

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bound on aij conversant with the total number of nodes that result in a stable consensus estimator. Consider a homogeneous communication structure, where it can be safely assumed that under fault free communication aij = amax. If the poles of such a networked system lie within the unit circle of the discrete pole-zero plot, amax represents a higher limit of arc weights of the communication graph. The magnitude of this amax determines the convergence rate of consensus-based observer. Practically, the value of amax is kept much smaller than this maximum value to guarantee the stability of the overall system. The consensus-based controls can converge a system that has dynamics much slower than the system dominant poles exhibit under this maximum value. If a faster response is required than the use of consensusbased controls only cannot guarantee stability. In comparison, The FIR based estimators construct the output from a finite input history. The “length” of the filter is composed of several unit time delays each representing one system node. For an FIR estimator of an order N, the convergence of the system to step input can be obtained in N steps. Such estimators can estimate system averages accurately in the presence of communication delays within very small time. The performance of these estimators is resilient to communication delays and therefore the overall performance of the proposed power-sharing controls is improved.

8.2. Frequency regulation This section compares the results of proposed multi-agents-based frequency restoration method while working with distributed averaging-based power-sharing and consensus-based power-sharing schemes. Fig. 8a shows the results of frequency restoration under the proposed power-sharing method, whereas Fig. 8c presents frequency restoration with consensus of observers method. A time delay is observed in the presence of link latencies. With the proposed method, system frequency restores to the desired value in lesser time duration as compared with the consensus-based method. 8.3. Voltage regulation This section compares the results of voltage regulation for the proposed method with consensus-based controls. Fig. 8b represents voltage restoration under the proposed method whereas, Fig. 8d shows the results of the consensus-based scheme. Both methods achieve voltage restoration to desired values. However, the consensus-based control exhibits more divergence in node voltages during the first two seconds following the startup transient. Whereas, distributed power averagingbased method achieves voltage restoration without significant node voltage deviations during this initial time period. For the proposed method, the converters in the system reach an agreement of corrective values for Pref.i, i and Vi , thereby, directing overall system power-sharing, frequency and voltage to desired values within finite time. However, for the consensus-based method the system frequency and voltage take more significant time to converge. Overall, the results of the proposed method show resilience towards communication delays wherein accurate power-sharing, voltage and frequency restoration are achieved within finite time.

8. Case simulation studies This section elaborates simulation studies undertaken in a MATLAB and Simulink environment developed for stability studies. Scenarios resulting in multiple communication link latencies are studied. To emulate communication latencies, a time delay of 10 ms to 5 s is introduced in the two links connecting DGU-6. The control algorithm subsequently strives to achieve equal power-sharing along with voltage and frequency restoration as demonstrated in Figs. 7 and 8. Compared to consensus-based controls, the distributed average estimation method performs better. Even with large communication delays, the powersharing between nodes remains effective. Whereas, in consensus-based methods, time delays lead to observable divergence in controlled parameters.

8.4. Reactive power sharing This section presents the results of reactive power-sharing between power injecting converter nodes. The reactive power sharing control directly competes with the voltage restoration controls. Therefore, a divergence in converter node voltages can be seen when reactive power is effectively shared. A suitable trade-off can be established between the two goals based on grid-specific requirements. Fig. 9a presents the results of reactive power sharing through the proposed controls. Fig. 9b shows the performance of the proposed controls under a load variation. Fig. 9c shows the corresponding node voltages where a divergence in node voltages is observed.

8.1. Active power sharing The effectiveness of the proposed active power-sharing method is evaluated against a consensus observer approach given as (23). Two sub-scenarios are considered for this experiment, in the first one, communication network between nodes form a complete ring digraph as represented in Fig. 2b. All nodes receive information from at least two neighboring nodes. No communication latencies are considered at this stage. Fig. 7a shows the results of power-sharing with the proposed method and Fig. 7b gives the results of power-sharing with consensusbased control for this scenario. The proposed method achieves powersharing more effectively as compared to the consensus-based method that exhibits measurable mismatch between active power injected by each node. The second scenario considers dual-link latencies leading to an isolated delay of information transmission directed towards and from DGU-6 as shown in Fig. 2c. Fig. 7c and d compare the results of power-sharing under the proposed averaging based method and consensus observer-based control for this scenario. A divergence in injected powers can be observed in the case of the consensus-based method. Whereas, the proposed method achieves power-sharing within a finite time.

8.5. Time varying delays in communication links and load variations This section studies the effect of time varying delays in communication links and system load variations. Dual-link latencies affecting communication links leading to DGU-6 i.e. a16, a61, a56 and a65 are considered. In terms of distributed control, this represents a plausible scenario for this study where one node is periodically isolated from neighboring network information. Furthermore, load variations are considered at three of the six system buses at the same instant to emulate a scenario where all load variations appear at the same time. The time variations introduced, delay the information signal by To(t), where To is a time-varying delay function as shown in Fig. 10a. The delayed signal may be represented by a generalized delay function y (t) = u(t − To(t)), where u(t − To(t)) represents the delayed input to the system and y(t) represents the output [48]. The variable delays here are of two types: td1 is a step delay spanning 1 s that starts at t = 4.15 s;

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Fig. 9. Performance of reactive power sharing under the proposed algorithm: (a) Without load variation (b) With load variation (c) voltage restoration results.

at t = 5.15 s the links are resumed. td2 is a ramp function starting at t = 7.5 s and reaching a maximum value at t = 10 s before abruptly falling to zero. Load variations of 0.33 p.u. are introduced at t = 5 s at busses 2, 3 and 6. At t = 10 s these additional loads are removed. Fig. 10b presents the results obtained with these tests using the proposed method whereas Fig. 10c presents the results obtained using conventional neighbor consensus-based control. Fig. 10d presents reactive power sharing results with the proposed method and Fig. 10e shows results of reactive power sharing using consensus-based controls. Fig. 10f and g compare results of frequency restoration using the proposed and consensus-based methods respectively. Fig. 10h and i compare results for voltage restoration. By comparison of these figures, it can be concluded that the proposed control method is more resilient to communication latencies. It effectively shares the active and reactive power between nodes and converges the MG system to nominal values of voltage and frequency in lesser time.

as references that are achieved through distributed averaging observers. the proposed error estimation method is resilient to errors resulting from delayed transmission of measurements and therefore results in a more robust control method that doesn’t diverge in presence of communication latencies as can be seen in Fig. 7a and c, Fig. 8a and b. Fig. 10a–i reflect that the proposed control method is more resilient to time-varying communication delays than consensus-based controls. The overall mechanism is resilient to the effects of latent and failing communication links and converges the system to stable states as observed from above discussed case studies. The behavior of these control schema is compared using miscellaneous parameters in Table 4. The proposed control strategy fares better than the conventional consensusbased controls on several accounts as shown. 9. Conclusion This work proposes a hierarchical control scheme for power-sharing between distributed generation units in an islanded AC micro-grid. Localized power estimators based on a novel distributed averaging method are developed to arrive at an estimate of instantaneous power injected through each node. These estimates are used as references for the droop power sharing control. A multi-agent-based secondary control layer for voltage and frequency regulation is implemented to observe and minimize deviations in key system parameters. The proposed method stabilizes the MG system operation during communication latencies in the communication network layer. A comprehensive discretetime mathematical small-signal model of the complete system is derived to analyze the performance of the proposed methodology. Moreover, case simulation studies are done to test the effectiveness of the proposed method. Comparison of the proposed method with conventional consensus-based control scheme shows that the proposed strategy adds

8.6. A comparison with Consensus-based control strategies This section concludes the comparison of the proposed strategy with existing conventional consensus-based strategies and evaluates its relative merits. Control schemes relying on multiagent consensus as described in [17,24,25,28,30], require an agreement of observers at distributed nodes to achieve a collaborated control. Such strategies are heavily dependent on the health of communication links to achieve convergence in system parameters. The consensus-based methods are integral methods that achieve consensus by collective error minimization. Communication latencies cause them to develop erroneous local estimates that magnify due to the integral effect of the looped integral controllers, and the system values diverge as can be seen in Fig. 7d. In comparison, the strategy proposed in this paper uses average estimates

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Fig. 10. Performance of the control schemes under delays and load variation: (a) Varying time delay function (b) Active power sharing with proposed method (c) Active power sharing with consensus-based method (d) Reactive power sharing with proposed method (e) Reactive power sharing with consensus-based method (f) Frequency restoration with proposed method (g) Frequency restoration with observer consensus of neighbour nodes (h) Voltage restoration with proposed method (i) Voltage restoration with observer consensus of neighbour nodes.

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Table 4 Comparison of Relative Merits of the Proposed Strategy with Conventional Strategies Under Communication Latencies. Parameters compared

Proposed method

Consensus Based Methods [17,24,25,28,30]

Active Power Convergence Reactive Power Convergence Voltage Restoration

Convergence achieved Convergence Achieved (trade off with Voltage restoration) Minimal variations observed that decay in small time span (trade-off with Reactive power sharing) Minimal variations that decay in small time span Small Small More

Partial- convergence Partial-Convergence (trade off with voltage restoration) Large Variations that persist Partial-Convergence (trade-off with Reactive power sharing) Large Variations observed that decay in longer time span Large Very Large Less

Frequency Restoration Frequency convergence time Voltage convergence time Resilience under time varying delays

resilience to the system against control errors caused by communication link latencies. The analytical proofs, stability analyses, and simulation studies reveal that the proposed method is sufficiently resilient to the effect of communication latencies in the communication and control network.

Acknowledgements The authors would like to thank the editor in chief, the associate editors and the reviewers for their worthy suggestions, time and effort in improving and finalizing this paper.

Declaration of Competing Interest The authors declared that there is no conflict of interest. Appendix A A.1. System matrices rLf Lf

0 rLf

0

ALCL =

Lf

1 Cf

0

0

1 Cf

0

0

0

ANETi =

B1Loadi =

0

rlinei Llinei 0

... ...

1 Lf

0

0

0

0

1

0

0

1

0

Lf

0

0

0

1 Lc

0

rLc Lc

0

1 Lc

0

BLCL2 =

0

0

1 LLoadi

. . .L

RLoadi LLoadi

1

AMG =

1 Lf

Loadi

. . .0

...

1 LLoadi

...

ANet = Diag [ ANet1 , ANet 2 .. ANetN ]2N × 2N 1 T Lf

0 1 Lf

BNETi =

RLoadi LLoadi

T

0 0 0 0

0 0 0 0

0

0

BLCL3 = [ Ilq

IlineQi IlineQi

B2Loadi =

ILoadQi ILoadDi

1 0 1 1 0 1 Minv = 0 0 0 0 , Mnet = 1 MLoad = 0 0 0 0 1 0 1 rN 2m × 2n 1 1 0 1 2m × 2s 1 2m × 2n Amg1 Binv RN MNet Binv RN MLoad Amg 2 ANet + B1Net RN MNet B1Net RN MLoad Amg1 = Ainv + Binv RN Minv Cinvc Amg 3 B1Load RN MNet Aload + B1Load RN MLoad

rN

Ainv1 + B1wcom Cinvw1 Ainv2 + B2wcom Cinvw2

.

Cinv = Diag [[Cinvc1 ] [Cinvc 2] . [CinvcN ]]2s × 13s Ainvi =

Biwcom = [ BPwcom 0 0 0 ]T13 × 1 CINVwi =

B2Net = [ B2Net1, B2Net 2 . B2NetN ]T 2N × 1 B1Net = [ B1Net1, B1Net 2 . B1NetN ]T 2N × 2m

Ioq ]T

Vod Ioq

0

Amg 2 = B1Net RN Minv Cinvc + B2Net Cinv Amg 3 = B1Load RN Minv Cinvc + B2Load Cinv rpki =

Ainv =

Ild Voq

B1NETi =

...

1 Llinei

0

0

1

...

. . .L

1

linei

Llinei

. . .0

0

...

1

...

Llinei

2 × 2m

2 × 2m

1

RN =

0

rLc Lc

0

0

0 0 0 0 0

0 0 0 0

Cf

ALoadi =

rlinei Llinei

LLoadi

1

0

0

0

1

Cf

BLCL1 =

1 Lf

{

1

2m × 2p

i

akk

Binv = [ Binv1 Binv 2 . BinvN ]T 13 × 2m

AinvN + BNwcom CinvwN 13× 13 APi 0 0 BPi BV 1i CPvi 0 0 BV 2i BC1i DV 1i CPvi BC1i CVi 0 BC1i DV 2i + BC 2i BLCL1i DC1i DV 1i CPvi+ BLCLli DCli CVi BLCLi CCi ALCLi + BLCL2i [TVi1 00] BLCL3i CPwi BLCLli (DC1i DV 2i + DC 2i )

[Cpw 0 0 0 ]1 × 13 i = 1 [0 0 0 0]1 × 13 i

1

1

}

13 × 13

where all entries of the matrices AMG, Ainvi, Binvi, Biwcom, CINVwi, CINVci represent sub17

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matrices describing the sub systems described in Section 5 of the paper A.2. Connectivity graph matrices

0 1 AG = 0 0 0 1

1 0 1 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 1

1 2 0 0 0 0 2 0 0 D = 0 0 2 G 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 L = 0 G 0 2

2 0 0 0

1

2

1

0 0 0

1

0

1

2 0 0

1

0 0

1

2 0

1

0 0 0

1

2

1

0 0 0

1

0

1

1

A.3. Admittance matrix

YbusMG

(Ys1 + Y17 + Y12) Y12 Y21 (Ys2 + Y23 + Y12) 0 Y32 (Ys3 + = 0 0 0 0 0 0 Y71 0

0 0 0 Y23 0 0 Y32 + Y34) Y34 0 Y43 (Ys4 + Y43 + Y45) Y45 0 Y54 (Ys5 + Y54 + Y56) 0 0 Y65 (Ys6 0 0 0

0 Y17 0 0 0 0 0 0 Y56 0 + Y65) 0 0 (Y71 + Ys7)

Here, inverter LC-L coupling admittance is represented as Ysi; line admittance between ith and jth nodes is represented by Yij; the current injected into the ith bus is denoted as Isi. A.4. Current voltage transformations

[ i 0DQ] = [T ]. [iodq] =

1].

[ vbdq] = [T

cos( ) sin( )

[ubDQ] =

sin( ) . [ iodq] + cos( )

Iod. cos( ) Iod. sin( )

cos( ) sin( ) . [ vbDQ ] + sin( ) cos( )

where the transformation matrix is [T ] =

Iod . sin( ) .[ Ioq cos( )

UbD sin( )

UbQ cos( )

UbD cos( )

UbQ sin( )

]

[

]

cos( ) sin( ) sin( ) cos( )

A.5. Inner control matrices

B v1 = 1 0 , B v 2 = 0 0 0 1 0 0

0

1

0

0 0 C = 1 0 0 V

0 0 KIVi 0 KPVi 0 , DV = , DV 2 = 0 KIVi 0 KPVi 0 0

KPVi n Cf

n Cf Fi 0 KPVi 0 Fi

KPCi KICi 0 KPCi 0 n Lf 0 0 0 0 1 0 0 0 0 0 BC1 = 1 0 , BC 2 = C = , DC1 = , DC 2 = the proportional and in0 1 0 1 0 0 0 0 C 0 KPCi 0 KICi KPCi 0 0 0 0 n Lf tegral gains of the voltage controller are given as KPVi and KIVi, output feed-back gain is denoted by Fi. the proportional and integral gains of the voltage controller are represented by KPCi and KICi. A.6. Power control matrices

0 AP = 0 0

mp 0

CPw = [ 0

mp 0 ], CPv =

c

0 0

, BPcom = c

0 0 0 0

0 0

1

0 0 , BP = 0 0 0 0

0

c Iod c Ioq

0

c Ioq c Iod

0

0

c Vod

c Ioq

c Voq

c Vod

nq 0

where steady state values of terms iod , ioq , vod , voq are represented by Iod , Ioq , Vod andVoq . The cut-off frequency for low pass filters used in the power calculator are defined as ci .

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