Synchronization of Dynamical Networks with a CommunicationInfrastructure: A Smart Grid Application

Jairo Giraldo, Eduardo Mojica-Nava, and Nicanor Quijano

Abstract— Motivated by the increasing interest in networkedmulti-agent systems and the wide number of applications indistributed control of smart grids, we address the problemof synchronization of microgrids. Two topologies are consid-ered: the physical topology that relates the interconnectionof distributed generators and loads, and the communicationtopology, which describes the information flow of the powersystem measurements. We propose a control strategy basedon the information flow and we show that there exists astrong relationship between both topologies in order to achievesynchronization. As a matter of fact, we show that our resultscan be extended when isolated group of nodes are taken intoaccount, and they can be connected or disconnected to the maingrid. Finally, the effects of time-varying sampling are analyzedusing some average passivity conditions, and sampling-timeindependence is demonstrated with the proposed controller

I. INTRODUCTION

The current power network is a nonlinear, complex, andlarge-scale system formed by the interconnection of severalnodes, such as loads and energy generators. However, in thelast few years the increasing need to provide reliable, high-quality electric power in an environmentally friendly andsustainable way has been of great interest in the researchcommunity, leading to the smart grid [1]. Smart grids arecharacterized by the inclusion of renewable resources intothe network and a communication infrastructure that allowsto make “smart” decisions based on a high amount ofinformation flow [2]. There exist several control needs inpower networks in order to assure quality of the electricityand to avoid faults or cascade failures. One of the mostimportant control goals is the preservation of synchroniza-tion between the whole power grid. Synchronization inAC systems consists on maintaining phase and frequencyof voltage and current equal among the power grid [3].Nevertheless, the inclusion of renewable resources, and thecapability of connecting and disconnecting to/from the grid(usually employed by microgrids in islanded mode [4])increases the possibility of asynchronous behavior, whichcan cause disturbances and oscillations that affect not onlythe generator, but also the stability of the power system[5]. The analysis of synchronization is commonly developedusing the well known swing equations of power systems,which are very useful for transient stability and small-signal

This work has been supported by CIFI 2011, Facultad de Ingenierıa,Universidad de los Andes, and Proyecto SILICE 3, Colciencias-Codensa.

J. Giraldo and N. Quijano are with Departamento de Inge-nierıa Electrica y Electronica, Universidad de los Andes, Colombia.{ja.giraldo908,nquijano}@uniandes.edu.co.E. Mojica-Nava is with Electrical and Electronics Department, NationalUniversity of Colombia. eamojican@unal.edu.co

stability analysis, modeling the interconnection of generatorsand their power energy exchange [6]. Furthermore, in [7] ithas been demonstrated the close relationship between swingequations and coupled oscillator models, specially with theKuramoto oscillator model, whose properties are very usefulfor the study of synchronization in power networks. Severalworks have focused on the development of synchronizationconditions for power systems based on the Kuramoto model.For instance, in [8] and [9] it has been shown that thephysical interconnection topology and power capacity ofthe transmission lines are of major importance in order toassure synchronization. Moreover, in [10] some conditionshave been established for less restrictive systems with energylosses and transfer susceptance using singular perturbationanalysis of the Kuramoto model. Still, synchronization de-pends on the connectivity of the physical interconnection,i.e., disconnected nodes are not considered, which is an openproblem that needs to be addressed.

The aim of this work is to develop new synchronizationconditions for power networks considering the communica-tion infrastructure presented in the smart grid. Few workshave included communications into the synchronization ofpower networks. In [11] [12], the communication infrastruc-ture plays an important role in control and synchronization.However, none of these works assume disconnected nodes,which is of great importance in the stability of smart grids.Our work leads to the possibility of synchronizing grids withisolated generators, microgrids, or even coupled microgridsthat can be connected or disconnected to the main grid atany time. Nevertheless, the inclusion of a communicationinfrastructure adds new challenges in control and stability[13], where communication constraints emerge, e.g., time-delays, packet losses, sampling and data rate, among others.We focus our attention on the sampling problem, wheredata transmitted between nodes is sampled, inducing somelimitations on the performance of the system [14]. Weextend our synchronization analysis under sampling usingthe concept called “average passivity” introduced in [15],and demonstrate that, with the proposed control strategy,synchronization is assured independently of the samplingperiod.

The paper is organized as follows. In Section II, the“smart grid” power balance is formulated using the swingequations based on phase and frequency changes, and somesynchronization criteria are introduced. The proposed controlstrategy using a communication infrastructure and synchro-nization conditions are presented in Section III, and itsextension considering sampling in Section IV. In Section V,

52nd IEEE Conference on Decision and ControlDecember 10-13, 2013. Florence, Italy

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simulation results are introduced and finally, some conclu-sions are drawn in Section VI.

Preliminaries and notationVectors and sets: Let C be the set of complex numbers,where jR ⊂ C represents only the imaginary part. Let usdefine υ ∈ N of N ordered increasing elements such thatυ = {1, . . . , N}. υ can be divided into m partions such thatυ = υ1∪υ2∪ . . .∪υm. The vector 0q is the vector of q zeroelements, and 1q as the vector of ones of size q.Graph theory: Let G =

{V, E ,AG

}represents an undirected

graph, where V = {1, . . . , N} is the set of nodes or vertices,and E = {(i, j)|i, j ∈ V} is the set of edges or adjacentnodes. The adjacency matrix AG = [aij ] is the symmetricmatrix N ×N , where aij = 1 if (i, j) are adjacent, aij = 0otherwise, and aii = 0 for all i ∈ V . For the ith node,the degree of a vertex di is the number of neighbors thatare adjacent to i, i.e., di =

∑Nj=1 aij . A sequence of edges

(i1, i2), (i2, i3), . . . , (ir−1, ir) is called a path from node i1to node ir. The graph G is said to be connected if for anyi, j ∈ V there is a path from i to j. The degree matrix isD = diag(d1, d2, . . . , dN ), and the Laplacian of G is definedas L = D−AG . Disconnected graphs can be divided into cconnected subgraphs G = G1 ∪ . . . ∪ Gc.Geometry of the n-torus: The set S1 is denoted as the unitcircle and any angle θ ∈ S1. |θ1 − θ2| is the distance(geodesic) between two angles θ1, θ2 ∈ S1. The sum ofn unit circles is called the n − torus, described by Tn =S1 × . . . × S1. For γ ∈ [0, 2π], the Arcn(γ) ⊂ Tn is theclosed set of angles θ = (θ1, . . . , θn), with the property thatthere exists an arc of length γ containing all θ1, . . . , θn.

II. POWER SYSTEMS SYNCHRONIZATION

One of the main goals in control of a power system issynchronization, where each node needs to work at the samefrequency and voltage in order to avoid failures and mal-functioning. Synchronization issues can be scaled to mediumand low voltage generators, such as distributed generators ina microgrid. Then, control strategies need to be developedin order to assure synchronization of distributed generators(e.g., renewable sources), loads, and synchronous generators.Furthermore, the complexity of the problem increases whenthe connection and disconnection of nodes are considered.Synchronization of these isolated nodes is important due tothe fact that, when nodes reconnect to the main grid, phaseand frequency need to be synchronized in order to avoidfailures.

A. Synchronization Model

Several works have considered the problem of synchro-nization of power systems from a point of view of synchro-nization of networks and oscillators, such as in [10] and [8],where the topology of the network plays a fundamental rolein assuring synchronization of synchronous generators.In this work, we will focus on the models introduced in [16],where the power grid model considers three types of nodes: i)conventional synchronous generators (υ1); ii) direct current(DC) power sources (υ2); and iii) load buses (υ3), where

each set υ1, υ2, υ3 is a partition of the the total orderedset described by υ = {1, . . . , N} = υ1 ∪ υ2 ∪ υ3, with|υ| = N be the number of nodes in the grid. The powerAC network is described by the purely inductive admittancematrix Y ∈ jRN×N , the nodal voltage magnitude Ei, andnodal voltage phase θi ∈ S1. The voltage magnitude isassumed to be constant, such that aij = |Ei||Ej ||Yij | is themaximum real power transfer between nodes i and j.The following model is based on the well known swingequations of the network reduced power system model [6],where generators and loads are described as phase oscillatorsaccording to [7]. The equations are derived from an energybalance between the elements connected to a node, and thetransmitted power. We assume that the voltage angles andthe rotor angles are the same, and transmission lines arelossless with zero conductance. Then, for the power networkwith N nodes whose physical connectivity is described by agraph Gp = {Vp, Ep,Ap}, with adjacency matrix Ap = [aij ],the swing equations for each type of node are described asfollows.i) For the first case, i.e., the conventional synchronousgenerator, the swing dynamics of each generator can bedescribed byMiθi +Diθi = Pm,i −

∑Nj=1 aij sin(θi − θj), i ∈ υ1

(1)where θi is the generator rotor angle, θi is the frequencyor speed of the generator, Pm,i is the mechanical powerinput, and Mi and Di are the inertia and damping coefficientsrespectively. In this work, frequency synchronization analysisis based on the first-order Kuramoto oscillator properties.Then, the inertia term Mi can be omitted due to the factthat it only modifies the convergence time to synchronizethe state but not its existence.ii) For the DC sources, it is assumed that each source (e.g.,solar panels or wind turbines), is connected to the AC grid us-ing a DC/AC inverter, which is equipped with a conventionaldroop-controller with a nominal power Pd,i > 0. The term1/Di > 0 is the droop-slope and the frequency deviationDiθi is proportional to the difference between the outputpower (

∑Nj=1 aij sin(θi − θj)) and the nominal power [17].

Then, the dynamics of the inverter can be described byDiθi = Pd,i −

∑Nj=1 aij sin(θi − θj), i ∈ υ2 (2)

iii) Finally, for the load buses case, each load consumesan amount of power PL,i, with a frequency-dependent termDiθi with Di > 0 that introduces the effects of frequencychanges in loads, such that

Diθi = −PL,i −∑Nj=1 aij sin(θi − θj), i ∈ υ3 (3)

Results can be easily extended for constant power loads, withDi = 0. However, for the sake of obtaining a more generalresult, we only consider frequency-dependant loads.

B. Synchronization Concepts for Power Systems

The analysis of synchronization in power networks hasbeen developed using the Kuramoto oscillator model [7][10]. It has been shown that synchronization properties ofKuramoto oscillators and the swing equations are equivalent.In [10], it was proven that the analysis of synchronous

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generators can be developed considering only the first orderdynamics, similar to a Kuramoto oscillator, omitting theinertia term. In this context, the phase balance dynamics arerewritten as follows

Diθi = Pi −∑N

j=1aij sin(θi − θj) (4)

where P = (Pm, Pd,−PL). Then, we introduce briefly somedefinitions regarding frequency and phase synchronizationbased on the models introduced above.

• Phase cohesiveness: For non-uniform Kuramoto oscil-lators, we refer to phase cohesiveness to the case wheneach pairwise distance |θi(t) − θj(t)| is bounded bya constant value γ ∈ (0, π], for all {i, j} ∈ E . Wedefine ∆(γ) :=

{θ ∈ TN : |θi − θj | ≤ γ

}where phase

cohesiveness is achieved if there exists a bounded valueγ such that θ(t) ∈ ∆(γ), for all t ≥ 0.

• Frequency Synchronization: Frequency synchroniza-tion is achieved when all frequencies θi(t) convergeto a common frequency ωsync ∈ R. This syn-chronization frequency is obtained by summing allequations omitting the second-order terms, such that∑Ni=1Diθi =

∑i∈υ1 Pm,i +

∑i∈υ2 Pd,i −

∑i∈υ3 PL,i −∑N

i=1

∑Nj=1 aij sin(θi − θj). The non symmetric property

of the sine function implies that the last term is zero.Since lim

t→∞θi(t) = ωsync is constant, we define P =

{Pm, Pd,−PL}, such that ωsync =∑Ni=1 Pi/

∑Ni=1Di.

We refer to synchronization of power networks to the casewhen frequency synchronization and phase cohesiveness areachieved, for some initial values θ(0) ∈ TN , θ(0) ∈ RN ,and P(0) ∈ RN .

III. COMMUNICATION INFRASTRUCTURE

The smart grid include a communication infrastructure inall the stages of the power system, from transmission tousers allowing the design of control strategies consideringinformation data flow. In real applications, we need to con-sider islanded elements, where synchronization needs to beassured such that, when the islanded nodes are reconnectedto the main power system, failures can be avoided and thestability of the network can be preserved. Therefore, the useof a communication may improve the behavior of the powersystem.

We consider two topologies or infrastructures, the phys-ical topology that describes the power system dynamics(as shown above), and the communication topology, whichdescribes how data from each node is transmitted. Bothtopologies can be described using graph theory, accordingto Figure 1. Then, data can be used in order to obtaincontrol signals to achieve synchronization. The next section

Isolated nodes

Fig. 1: The physical topology (circles) and the communicationtopology (squares). Both graphs can be different.

describes the close relationship between both topologies fora control strategy based on consensus.

A. Control Strategy and the Combination of Both Topologies

To design a control strategy for synchronization, it is nec-essary to collect information from some neighbors consistingof phasor measurement units (PMUs) or sensors, such thatlocal controllers integrated to some nodes (except for theload buses, which are not controlled) use the informationto calculate a control signal Pi ∈ R. The control signalcan be interpreted as power injection for positive Pi orpower absorption for negative values of Pi, which is executedusing storage devices (e.g., batteries) that can absorb orinject power to the generators buses [18]. Then, in Equations(1) and (2), Pm,i = P ∗m,i + Pi, for i ∈ υ1, and Pd,i =P ∗d,i +Pi for υ2, where P ∗m,i and P ∗d,i are positive constantscorresponding to the injected power in steady state operationand the constant nominal power respectively. We proposea control strategy that depends on the information of theneighbors of each node. Neighbors can be related usingthe adjacency matrix Bc = [bij ]. Therefore, the controlmethodology is given by

Pi = Ki

∑Nj=1 bij

(ˆθj − θi

), for some i ∈ υ (5)

where bij is a binary number, where 1 indicates that there ex-ists information flow from i to j, and zero otherwise. Ki ≥ 0is a positive constant value in Watts, which relates the amountof power injected/absorbed with the sum of the frequencydifferences. Ki=0, for load buses, i.e., for all i ∈ υ3 . Theterm ˆ

θj corresponds to the frequency data received from thejth neighbor according to the communication infrastructure.This data can be delayed or sampled, and it may presentpacket losses, noises, changes in communication topology,among others.

B. Synchronization Condition

The inclusion of a communication infrastructure addssome advantages in the synchronization criteria and improvessynchronizability and stability. Next, conditions for phasecohesiveness and frequency synchronization are introducedshowing the relationship between both physical and com-munication topologies, assuming that the information flowthrough the communication links is exact, i.e., ˆ

jθ = θj , forall j ∈ υ.

Theorem 1: Phase Cohesiveness: Consider the dynamicmodel of the power network based on Equations (1)-(3), witha physical topology described by an undirected and weightedgraph Gp(Vp, εp,Ap) with degree dAp

i , and a communicationtopology described by an undirected graph Gc(Vc, εc,Bc)with degree dBc

i , which transmit data without communicationconstraints, i.e., ˆ

θj = θj for all j ∈ υ. Assume thatfrequency synchronization is achieved for initial phase anglesθ(0) ∈ ∆(γ), such that θi = ωsync, and |θi(0)− θj(0)| ≤ γfor all i, j ∈ υ. We can also define P∗ = {P ∗m, P ∗d }. If inEquation (5)• Ki ≥

(P ∗i + d

Ap

i sin(γ))/(γdBci ) for i ∈ υ1 ∪ υ2, and

• |PL,i| ≤ dAp

i sin(γ), for all i ∈ υ3,

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then phase cohesiveness is assured 1.Phase cohesiveness can be interpreted as a condition

where the power exchange between nodes is bounded bylimiting the phase differences. Then, the selection of Ki

depends on the desired γ, and the steady state constantpower P ∗i . However, if we have several subgraphs of Gp, i.e.,isolated areas, at least one element of each area must haveinjecting/storing capabilities. In the following theorem, wewill obtain other conditions for Ki, that implies positiveness.

Remark 3.1: The maximum power delivery is obtainedwhen γ = π/2. Here, the amount of power PL that theload buses can absorb is maximum. Besides, the energyexchanged is maximized because sin(γ) is maximum. Addi-tionally, we can notice that conditions for phase cohesivenessare less restrictive than conditions in [16], where |P ∗i | ≤dAp

i sin(γ), for all i ∈ υ, and we can select γ ∈ (0, π).Now, we introduce some requirements for frequency syn-

chronization based on Laplacian properties.Theorem 2: Frequency Synchronization: Consider the

power network problem of N nodes described by (1)-(3).The initial phase condition is θ(0) ∈ ∆(γ). The physicaltopology is described by the undirected and weighted graphGp(Vp, εp,Ap) and the communication topology is describedby the undirected graph Gc(Vc, εc,Bc). The Laplacian rep-resentations of each graph is LA and LB, respectively. Weassume that K = diag(K,0|υ3|) for K = {Ki} chosenaccording to Theorem 1, and D = diag(D1, . . . , DN ).Assume that there exists a Laplacian given by

L = LA + KLB (6)whose eigenvalues are λ(L) = {λ1, . . . , λN}. If λ1 = 0 isthe only zero eigenvalue, and λi > 0 for i = 2, . . . , N ,then all frequencies asymptotically converge to ωsync =∑Ni=1 Pi/

∑Ni=1Di.

The proof is based on the frequency dynamics that canbe obtained by differentiating the model in Equation (4)including the controller dynamics in (5) such that

d

dtDθ = −

(KLB + LA(t)

)θ (7)

which is a consensus problem, where LA(t) ≤ LA. Then,the consensus criteria of the eigenvalues of the Laplacian Lneed to be assured.

Remark 3.2: In [10] it has been demonstrated that if anypair |θi(0) − θj(0)| ≤ γ ∈ [0, π/2] for each i, j ∈ υ,frequency synchronization is assured because LA(t) is al-ways positive. However, our condition is less restrictive and,because synchronization is achieved even when γ > π/2, aslong as Ki satisfies the conditions in Theorem 1.

IV. EFFECTS OF TIME-VARYING SAMPLING

As it was mentioned above, the inclusion of a communi-cation network allows synchronization of isolated nodes andmakes the synchronization conditions less restrictive. How-ever, communication networks also induce some constraintsthat may provoke undesired behavior or even instabilityof a system [19]. In this section, we focus on the effects

1The proofs of our theoretical results are not included due to lack ofspace.

of sampling, and we show that for the control techniquepresented in (5), and for homogenous and time-varyingsampling periods, synchronization is always achieved.

Consider the swing equations in (1)-(3) compacted in (4),and the control strategy in Section III. Frequency dynamicscan be obtained analyzing the frequency differences. How-ever, in this case, we assume that ˆ

θj corresponds to thefrequency data received from the jth neighbor and it is thesampled representation of θj . Thus, the term KLB is dividedinto KD(Bc)−KBc such that

d

dtDθ = −

(KD(Bc) + LA(t)

)︸ ︷︷ ︸

A

θ + KBc︸︷︷︸B

ˆθ (8)

where D(Bc) is the degree matrix of Bc, Bc is the adjacencymatrix that describes the communication topology, and ˆ

θis the vector of data received from all nodes. If samplingis not consider, Equation (8) is equal to (7). Sampling isdeveloped using zero-order holder with a sampling period δ.The ZOH provokes the input of each node to be constant fort ∈ [kδ, (k + 1)δ]. Then, the input and output of the wholesystem is described by y(t) = θ, and u(t) = ˆθ respectivelyand the state representation can be given by

Σ =

{ddtDθ = Aθ +Bu(t)

y(t) = θ(9)

Now, we introduce the following definitions adapted from[15], that will illustrate the stability of the power network in-dependent of sampling-period for systems with direct input-output links.- Passivity of Sampled-Data SystemsLet us consider a continuous-time system of the form

xc(t) = f(xc(t)) + g(xc(t))uc(t)

yc(t) = h(xc(t), uc(t)) (10)Definition 4.1: The system in (10) is passive if there exists

a storage function V (xc(t)) > 0, and V (0) = 0 such thatV (xc(t))− V (xc(0)) ≤

∫ t0yc(τ)uc(τ)dτ .

Definition 4.2: For the continuous-time system represen-tation above, assume that the input depends on sampled-data with a ZOH, such that uc is constant in the intervalt ∈ [δk, δ(k+ 1)], where δ is the sampling period. Passivityof the sampled-data representation reduces to passivity ofthe continuous-time system in each time interval where u∗cis constant, xc(0) = xc(k), and xc(t) = xc(k+ 1), such that

V (xc((k + 1)))− V (xc(k)) ≤(∫ δ

0

yc(τ)dτ

)u∗c (11)

The state xc(k) corresponds to the state of the discrete-timerepresentation and xc(k + 1) = eδ(f+u

∗c(k)g)xc(k) is the

solution for a constant u∗c(k). Then, the term related to theoutput yc(t) = h(xc(t), uc(t)) in Equation (11) is describedby ∫ δ

0

h(xc(τ), uc)dτ =

∫ δ

0

eτ(f+u∗c (k)g)h(xc(k), u∗c(k))dτ.

It is evident that the discrete-time output equivalent in theright-hand depends on the input u∗c(k) at each instant k.Hence, the following definition adapted from [15] is derived.

Definition 4.3: Passivity with input/output link: Pas-sivity (average passivity) of the sampled-data representationis preserved if there exists an input/output link, such that theoutput depends on the input. If there is not an input/outputlink, passivity is lost, and stability criteria cannot be defined.

The following theorem establishes conditions forsampling-independence of the synchronization problem

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based on the ideas introduced in Definitions 4.1-4.3.Theorem 3: Synchronization under sampling: Assume

the sampled data synchronization problem in (9), for con-stant input u∗(t) in the interval t ∈ [δk, δ(k + 1)] andoutput y(t) = x(t). If its continuous-time representationwithout sampling is strictly passive with storage functionV (x(t)) = 1

2x(t)>D>QDx(t) and y(t) = 12 (x(t) + u(t)),

then, its sampled-data representation is also strictly passiveindependent of the sampled period and synchronization isasymptotically achieved.

The proof is based on the analysis of the non-sampled casewhere the Laplacian L(t) = A+B. As the graph is directed,we need to define a diagonal matrix Q = diag{ri/si},where for ri > 0 and si > 0 such that L>(t)r = 0 andL(t)s = 0 [20]. Then, consensus of the time-varying directedgraph is achieved if the system is passive with respect toV (x(t)) = 1

2x(t)>D>QDx(t). Besides, as the output hasa direct input/output link, according to Definition 4.3 thesampled-data representation is passive and synchronizationis assured independent of the sampling period.

The aforementioned theorem shows that synchronizationis preserved using the communication infrastructure witha sampling period δ and it is independent of δ. The onlycondition lies in the eigenvalues of L(t) and are equivalentto conditions in Theorem 2. So, even if δ is time-varying,synchronization is preserved.

V. SIMULATION RESULTS

In this section, the modeling ideas presented in the previ-ous section are illustrated with the 9-bus system of Figure2 which is an adaption of the 3-machine, 9-bus model ofthe Western System Coordinating Council (WSCC 9-Bus)[6]. We consider two synchronous generators (nodes 1,2),three load buses (nodes 5,6,8), and one DC sources witha DC/AC inverter (e.g., high voltage solar panel in node3). We assume that the synchronous generators and theDC-source posses energy storage devices that aborb/injectpower. We assume that the initial conditions are boundedby θ(0) ∈ ∆(γ), for γ = π/2 The constant powersinjected to buses are Pm = (0.8, 1.63) p.u., Pd = 0.85p.u., and PL = (1, 0.9, 0.8)p.u. The admittance matrix isformed only by susceptance values neglecting the effects ofconductances (i.e., the line transmission are lossless), andthe susceptance of transformers is zero. We assume that the

G G

G

1

2 7 8 9 3

65

4

Fig. 2: WSCC 9-buses scheme.generator associated to node 1 and the load in node 5 areisolated, and after 5 seconds, they reconnect to the maingrid. When there is not communication nor control, Figure

3 depicts the frequency, phase differences and power of allgenerators and loads in the system when |P ∗i | = |Pi| ≤dAp

i sin(γ) is satisfied for all i. Synchronization is achievedindependently for the two formed subsystems before theinterconnection, and they synchronize to their own ωnsync, forn = 1, 2. However, when the interconnection is introducedafter 5 seconds, they cannot longer maintain synchronization,provoking unstable behaviors. On the contrary, Figure 4illustrates that, with the proposed control strategy and thecommunication topology that transmit sampled data with asampling period of δ = 0.05s, synchronization is achievedeven when switching is present. The selection of δ is dictatedby the typical PMU’s sampling time, and they sent data at thesame time. In Figure 4, dashed lines of frequency differences(top) are not inside the set ∆(γ) before the interconnection,because they illustrate the phase differences between nodes1 and 5, with respect to other nodes that are not physicallyrelated. However, when they connect to the main network,phase cohesiveness is preserved, for phase differences |θi −θj | ≤ γ + 2nijπ, where nij ∈ N. Figure 4 (Bottom),illustrate the power injection/absorbsion in the buses relatedto the synchronous generators and the DC source. Beforethe reconnection, the control signal indicates that a powerabsorbtion is needed in nodes 1, 2 in order to maintain thepower balance and frequency synchronization. On the otherhand, the DC source needed an extra power injection at thebeginning due to the impossibility of delivering the necessarypower demanded by the load in node 8. However, after onesecond, it starts absorbing power. When all isolated elementsreconnect to the main grid after 5 seconds, all control devicesneed to absorb the surplus of energy in the network.

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

Time (sec)

Fre

quen

cy (θ

i)

−20

−10

0

10

20

30

40

(θi −

θj)

Syn. Gen1

Syn. Gen2

Load Bus1

Load Bus2

Load Bus3

DC1

Fig. 3: Phase differences and frequency on each node of theWSCC 9-buses system where nodes 1 and 5 are isolated, without acommunication infrastructure. After 5 seconds, the isolated nodesare reconnected.

Simulations illustrate the usefulness of the proposed con-trol strategy and the possibility of achieving synchronizationeven when changes in the topology appears. For instance,loads may change or faults can be detected such that reclosersor switches activate, changing the physical interconnectionof nodes. Additionally, if we consider several microgridsthat can exchange energy between them, then the systemneeds to be robust enough to be synchronized with topo-logical changes. In this context, synchronization is assuredif conditions in Theorems 1 and 2 are satisfied. Then,

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−20

−15

−10

−5

0

5

Fre

quen

cy (θ

i)

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

Time (sec)

Pi

−10

−5

0

5

10

(θi −

θj)

Syn. Gen1

Syn. Gen2

Load Bus1

Load Bus2

Load Bus3

DC1

Fig. 4: Phase differences, frequency and powers on each node ofthe WSCC 9-buses system where nodes 1 and 5 are isolated witha communication infrastructure and power injection control. After5 seconds, the isolated nodes are reconnected.

disconnected areas can synchronize if frequency measure-ments exist. For the case when the isolated zones receivefrequency information from at least one non-isolated node(from a node of another area or microgrid), the whole systemsynchronizes, according to conditions in Theorem 2. Thisis a very important property because it allows to designrobust strategies to topological changes in a microgrid, with alow connected communication infrastructure that minimizescosts. In this context, a very important design parameter isthe sampling period. Small sampling times imply expensivecommunication devices. However, the property of sampling-time independence implies a decrease in the use of com-munication resources and low costs communication infras-tructures. Nevertheless, larger sampling periods mean slowconvergence time, which is not desirable in power systems,because of the possibility of cascade failures. Therefore, thedesign of a communication infrastructure combines the needof lower costs with sufficiently fast communications andconvergence time. Conditions in this work are easily scalableto large scale systems, with an increase of the communicationcapacities.

VI. CONCLUSIONS AND FUTURE WORK

We have investigated the synchronization problem ofpower systems based on the smart grid architecture thatincludes the communication infrastructure. We have basedour analysis on the reduced swing equations and we haveestablished some conditions that are less restrictive than inprevious works, using a novel control strategy. The proposedcontrol technique allows synchronization even when nodes ofnodes are disconnected. These results can be easily extendedto the study of synchronization of coupled microgrids thatcan work in isolated or connected mode. Additionally, wehave shown that the synchronization model with sampled-data information is independent of the sampling period,which is very useful for the integration of real applicationswhere communication constraints are present. Future workwill consider other communication constraints such as time-delays and packet losses, and the inclusion of more realmodels of the communication infrastructure such as wire-less communications, PLC, and time-varying communicationtopology. Besides, the extension for more accurate models

with conductances effects and without the assumption ofconstant voltages needs to be developed.

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