Tracking of Kuramoto Oscillators with Input Saturation andApplications in Smart Grids

Jairo Giraldo, Eduardo Mojica-Nava, and Nicanor Quijano

Abstract— The Kuramoto oscillator has been widelystudied because it can model biological, social, chemical,and engineering problems. Conditions for frequency syn-chronization of a network of Kuramoto oscillators havebeen well established, but it depends on the connectivityof the network and on the natural frequencies. In thiswork we propose a consensus-based control strategy thatforces the network to follow a virtual agent with constantfrequency. Besides, we consider that the control input isconstrained and we establish the limits of the set of inputsthat assure frequency tracking. The results are extended tothe smart grid case, where the nodes of a power networkare modeled as Kuramoto oscillators that synchronize to adesired frequency using the injection/absorbsion of poweraccording to a consensus based control. We show that ourresults are feasible when isolated group of nodes are takeninto account, and they can be connected or disconnectedto the main grid. Finally, the effects of time-varyingsampling are analyzed and sampling period independenceis demonstrated.

I. INTRODUCTION

Synchronization phenomena have been extensivelystudied in the last century due to the possibility ofmodeling the coupled dynamics of physical, social, bio-logical, and chemical systems [1]. Applications in engi-neering include networks of interconnected oscillators[2], and clock synchronization in computer networks[3], just to name a few. In [4], the well known Ku-ramoto oscillator dynamics were introduced. The richdynamical behavior of the Kuramoto equation illustratesthe competition between each agent, that tends to alignwith its natural frequency. The attention of the researchcommunity has been given to the self-synchronizingproperties of the network of Kuramoto oscillators andin the definition of the topological conditions that assurefrequency synchronization [5] However, there are twoopen research areas that have not been addressed inliterature: i) the special case when the network topologythat describes the coupled interactions are disconnecteddue to environmental limitations or physical failures thatprovoke that an agent or group agents become isolated;

This work has been supported by Proyecto SILICE 3, Colciencias-Codensa.

J. Giraldo and N. Quijano are with Departamento de Inge-niera Elctrica y Electrnica, 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

and ii) some applications require that the network ofoscillators oscillates to a desired frequency, according toa frequency reference.

In this work, we include into the Kuramoto modela control input that forces the network of oscillatorsto oscillate to a desired reference even when the graphdescribing the coupling dynamics is disconnected. In [6]authors introduced a control strategy to force complexnetworks to achieve a desired reference state, when theindividual dynamics of each oscillator depend only ontheir own states. In our case, each Kuramoto oscillatordepends on its interaction with other oscillators, de-scribed by a physical topology. Additionally, we assumethat the input signal is bounded and we establish the setof inputs that assure synchronization and tracking.

Several engineering applications of the Kuramotomodel have been studied in [7] [8], where the authorshave exploited the similarities of a power network witha network of Kuramoto oscillators, and they have de-rived the network requirements to achieve frequencysynchronization. Nevertheless, we have extended theseresults taking advantage of the smart grid architecturethat combines a communication infrastructure with thepower network [9]. Few works have used the commu-nications capacity of the smart grid to deal with thesynchronization problem. [10], [11] However, all of themrequire connectivity of the power network, which is notpossible when we consider connection or disconnectionof microgrids. Then, using the proposed control strategywe can determine the amount of power that needs tobe injected/absorbed to some nodes, even when some ofthem are disconnected. On the other hand, we focus ouranalysis on the effects of sampling in the synchronizationproblem. It has been shown that sampling may provokethe system to get unstable [12]. Then, we prove that withthe proposed strategy, tracking and synchronization areachieved independent of the sampling period.

The paper is organized as follows. Section II in-troduces the proposed Kuramoto model that follows aconstant reference including input saturation. The appli-cation to smart grids are presented in Section III wheresampling independence is shown. Section IV illustratesthe effectiveness of our approach for a modified IEEE30 bus system, and finally some conclusions and futuredirections are drawn in Section V.

Preliminaries and notationLet C be the set of complex numbers, where jR ⊂ Crepresents only the imaginary part. The vector 0q is thevector of q zero elements, and 1q as the vector of onesof size q. Graph theory: Let G =

{V, E ,AG

}represents

a graph, where V = {1, 2, . . . , N} is the set of nodesor vertices, and E = {(i, j)|i, j ∈ V} is the set of pairscalled edges. If a pair (i, j) ∈ E , then i, j are said tobe adjacent. The adjacency matrix AG = [aij ] is thesymmetric matrix N × N , where aij = 1 if (i, j) areadjacent, aij = 0 otherwise. For the ith node, the in-degree of a vertex di is the number of neighbors whosedirection is heading to node i. A sequence of edges(i1, i2), (i2, i3), . . . , (ir−1, ir) is called a path from nodei1 to node ir. The graph G is said to be connected if forany i, j ∈ V there is a path from i to j. The degreematrix is D = diag(d1, d2, . . . , dN ), and the Laplacianof G is defined as L = D−AG . Disconnected graphs canbe divided into c connected subgraphs G = G1∪ . . .∪Gc.Geometry of the n-torus: The set S1 is denoted as theunit circle and any angle θ ∈ S1. |θ1−θ2| is the distance(geodesic) between two angles θ1, θ2 ∈ S1. The sumof n unit circles is called the n − torus, described byTn = S1 × . . . × S1. For γ ∈ [0, 2π], Arcn(γ) ⊂ Tnis the closed set of angles θ = (θ1, . . . , θn), with theproperty that there exists an arc of length γ containingall θ1, . . . , θn.

II. TRACKING MODEL OF THE KURAMOTOOSCILLATOR

The Kuramoto equation has been widely used tomodel the dynamics of N coupled oscillators (we alsocall them agents or nodes), that compete between themin order to force each of its neighbors to align with itsnatural frequency ωi, for i = 1, . . . , N . For networks ofoscillators, couplings are described by a graph Gp(V, Ep)with an adjacency matrix Ap, leading to the extendedKuramoto model Diθi = ωi−

∑Nj=1 aij sin(θi − θj) for

all i ∈ V , where Di > 0 is a damping or viscous coeffi-cient that introduces multiple time-constants, and aij arethe elements of the adjacency matrix Ap that describesthe physical interconnection of all the oscillators [13].Therefore, the following definitions are introduced.

Phase cohesiveness: For a solution θ =[θ1, . . . , θN ] ,>, phase cohesiveness is achieved ifeach pairwise distance |θi(t) − θj(t)| is bounded by aconstant value γ ∈ (0, π], for all {i, j} ∈ E . Then, wedefine ∆(γ) :=

{θ ∈ TN : |θi − θj | ≤ γ

}, such that

θ(t) ∈ ∆(γ), for all t ≥ 0.Frequency Synchronization: Frequency synchro-

nization is achieved when all frequencies θi(t) convergeto a common frequency ωsync ∈ R, which typically isgiven by ωsync =

∑Ni=1 ωi∑Ni=1Di

.

Then, we refer to synchronization of the coupled oscil-lators when phase cohesiveness and frequency synchro-nization are assured.

Researchers have focused on finding the conditionsto achieve frequency synchronization and phase co-hesiveness (also called self-organized synchronization)depending on the network topology and the naturalfrequencies of each oscillator. Some applications requirethat the network of oscillators synchronizes to a desiredfrequency maintaining phase cohesiveness even whensome nodes are not coupled or they are isolated.However, in literature it does not exist any techniquethat allows a network of Kuramoto oscillators to followa frequency reference θr ∈ R, such that ωsync = θr, andit remains as an open field. In this contex, we propose amodel that includes an input ui that enforces the networkto track a frequency reference. The model is as follows

Diθi = ωi −∑N

j=1aij sin(θi − θj) + ui. (1)

This input ui can be associated to some extra informationthat each agent possesses about the speed of oscillationof some neighbors. Under this input assumption, wepropose a consensus-based controller for synchronizationand frequency tracking.

A. Following a Constant Reference

The proposed controller is based on the consensusalgorithm that linearly combines the information ofsome neighbors according to a communication topology[14]. Then, we assume that each oscillator is able tomeasure its frequency, and share that information tosome neighbors. The information flow of a network ofN oscillators is described by a directed graph Gc(V, Ec)with an adjacency matrix Bc, with elements bij = 1if there exists an exchange of information between i, j,and bij = 0 otherwise. Therefore, as the objective is tofollow a constant reference, we include a virtual agentθN+1 = θr with linear phase θN+1 = θrt, where θr isthe constant slope. This agent transmits its informationto other agents, forcing the network to follow its constantstate [15]. Then, the controller of the ith agent isui = −Ki

∑N

j=1bij(θi − θj)−Kibi(N+1)(θi−θr) (2)

where bi(N+1) = 1 if the ith agent is con-nected to the virtual agent and bi(N+1) = 0 other-wise. We can define the augmented graph represen-tation GN+1

c (VN+1, EN+1c ), with Laplacian LN+1

B ∈R(N+1)×(N+1), and with an adjacency matrix BN+1

c ∈R(N+1)×(N+1), such that the controller matrix formis described as ¯u = −KLN+1

B¯θ(t), for ¯

θ(t) =[θ1, . . . , θN , θ

r], K = diag(K1, . . . ,KN , 0), and ¯u =[u1, . . . , uN , 0]. The following theorems introduce theconditions to achieve frequency synchronization andphase cohesiveness based on the physical and the com-munication topologies, and the selection of the constant

Ki for all i ∈ V .Theorem 1: Frequency Synchronization: Consider

the network of N coupled oscillators described by (1)and (2). Assume that the initial phase condition isθ(0) ∈ ∆(γ). The physical topology is described bythe undirected and weighted graph Gp(V, εp,Ap) andthe communication topology is described by the graphGc(V, εc,Bc). The augmented Laplacian representationsof each graph is LN+1

A and LN+1B , respectively. We

assume that D = diag(D1, . . . , DN , 0) and K =diag(Ki, . . . ,KN , 0), and we define the Laplacian L =LN+1A + KLN+1

B ≥ 0 whose eigenvalues are λ(L) ={λ1, . . . , λN+1}. If λ1 = 0 is the only zero eigenvalue,and λi > 0 for i = 2, . . . , N + 1, then all frequenciesconverge to ωsync = θr.Sketch of the proof: The frequency dynamics can beobtained by differentiating the model in Equation (1)including the controller dynamics in (2) such that

d

dtD¯θ = −

(KLN+1

B + LN+1A (t)

)¯θ (3)

where LN+1A (t) is the Laplacian of the time-varying

graph described by the connectivity matrix AN+1p , whose

elements are given by aij = aijcos(θi − θj), for alli, j ∈ V , and ai(N+1) = 0 for all i ∈ VN+1. As cos(θi−θj) ∈ [−1, 1], L(t) ≤ L for L = KLN+1

B +LN+1A , then,

the synchronization condition reduces to the analysis ofthe eigenvalues λ(L). If the augmented total graph isconnected, all agents synchronizes to θr �

Note that frequency synchronization depends on theconnectivity of the coupled graphs. Then, if the graphdescribing the physical interconnection is disconnected,the disconnection is compensated by the communicationgraph. As a consequence, both topologies do not needto be connected, but the resulting graph does. However,there are restrictions regarding the selection of Ki thatare addressed in the next theorem.

Theorem 2: Phase Cohesiveness: Consider the dy-namic model in Equations (1) and (2), with a physi-cal topology described by an undirected and weightedgraph Gp(V, εp,Ap) with degree dAp

i , and an augmentedcommunication topology described by a directed graphGN+1c (VN+1, εN+1

c ) with an adjacency matrix BN+1c of

degree dBN+1ci , for all i ∈ V . If there exists a solution

θ ∈ ∆(γ), for γ ∈ [0, π] and initial phase anglesθ(0) ∈ ∆(γ), then the following condition hold:

Ki ≥|Diθ

r−ωi|−dApi sin(γ)

dBN+1c

i γ, for all i ∈ V (4)

Proof: First, integrating the controller dynamics, we ob-tain that ui = −Ki

∑N+1j=1 bij(θi − θj). The equilibrium

solution is achieved when θi = θr for all i ∈ V . Then,Equation (1) can be rewritten asDiθ

r−ωi = −Ki

∑N+1

j=1bij(θi − θj)−

∑N

j=1aij sin(θi − θj).

(5)Since sin(θi − θj) ∈ [− sin(γ),+ sin(γ)] and ui ∈

[−KidBN+1

ci γ,+Kid

BN+1ci γ, ] for θ ∈ ∆(γ), then

|Diθr − ωi| ≤ d

Ap

i sin(γ) + KidBN+1

ci γ. As a conse-

quence, Equation (5) has no solution if condition in (4)is not satisfied. �

Remark 2.1: Note that not necessarily all agents maypossess a control input. Then, Ki = 0 for a subset υ ⊂ Vsuch that phase cohesiveness is preserved if conditionsin Theorem 1 are satisfied and∣∣∣Diθ

r − ωi

∣∣∣ ≤ dAp

i sin(γ), for all i ∈ υ. (6)

Theorems 1 and 2 depend on the description of bothtopologies, the selection of Ki, and the natural frequen-cies. However, the control input may be constrained dueto environmental or physical limitations. Then, the nextsection introduces some design parameters for the inputui.

B. Tracking with Input Constraints

As we mentioned before, not all nodes need a controlinput, such that we can divide the set of vertex V intotwo sets: i) nodes without control υ = {i ∈ V|Ki = 0},and ii) agents with control inputs υc = {i ∈ V|Ki > 0},where V = υ ∪ υc. A good way of approximating inputconstraints is to include a saturation function sat(ui).Saturation introduces nonlinearities to the system wherewe can define two regions for ui: the region of saturationRsi , where ui = umini (umaxi ) if ui ≥ (≤)umini (umaxi )and the region of linearity RLi , where umini ≤ ui ≤umaxi . When saturation is taken into account, the Ku-ramoto model with control input becomes

Diθi = ωi −∑N

j=1aij sin(θi − θj) + sat(ui) (7)

for i ∈ υc. Differentiating Equation (7), we obtain thatd

dtDiθi =

−∑N

j=1 aij cos(θi − θj)(θi − θj

)+ ui, if ui ∈ RL

i

−∑N

j=1 aij cos(θi − θj)(θi − θj

), otherwise.

(8)

According to Equation (8), if an input ui goes fromRLi to Rsi , the dynamics will depend on the non-controlled system, such that ui = 0. As a matter offact, if all inputs are saturated, ui = 0 for all i ∈ υcand the Laplacian in Equation (3) will be given byL = LN+1

A (t) provoking frequencies to synchronize toωsync =

∑Ni=1 ωi/

∑Ni=1Di. Therefore, we need to find

the boundaries umini , umaxi for all i ∈ υc such that inputsremain in the region of linearity ui ∈ RLi as follows.

Theorem 3: Let us consider the Kuramoto model inEquation (1), with control input in (2). The initial phaseangles are θ(0) ∈ ∆(γ). Assume that the networksatisfies the conditions in Theorems 1 and 2. If theconstraints of the control input are given by

umaxi ≥ Diθ

r − ωi + dAi sin(γ)

umini ≤ Diθ

r − ωi − dAi sin(γ) (9)for all i ∈ υc, then the control input is never saturated

and frequency tracking is maintained.Proof:From Equation (2) we obtain that ui = Diθi − ωi +∑Nj=1 aij sin(θi − θj). As phase cohesiveness is main-

tained such that θ(t) ∈ ∆(γ) for all t, then we needto assure that the input ui is such that the equilibrium

¯θ = θr1N+1 is attained. Then, as sin(θi − θj) ∈[− sin(γ),+ sin(γ)], the maximum and the minimuminput values for all i ∈ υc, are described by Equation(9). �

The following lemmas bring up some important de-sign characteristics of the controllers and the physicaltopology.

Lemma 2.1: If the graph G(V, E ,Ap) is connected,tracking is possible only if at least one node possessescontrol capabilities without constraints, and it is con-nected to the virtual agent such that input remains inthe region of linearity umini ≤ ui ≤ umaxi , whilethe frequency reference is attained. Nevertheless, it isnecessary that condition in (6) is satisfied for thoseagents whose controllers can be saturated.

Lemma 2.2: Assume that the graph G(V, E ,Ap) isdisconnected forming g subgraphs G1, . . . ,Gg . Then, ifat least one node in each subgraph possesses unboundedcontrol input and each of them is connected to the virtualagent, such that conditions in Theorems 1 and 2 aresatisfied, synchronization is achieved and ωsync = θr.However, condition in Equation (6) must hold for agentswhose controllers can be saturated.The aforementioned lemmas give some requirements forinput saturation that can be used to design the number ofcontrollers, the communication topology, and the inputconstraints. However, there exists a tradeoff betweenthe convergence time and the design parameters of thecontroller.

Next, we will adapt our results for the smart grid case,where we have a set of power generators, loads, and DC-sources with inverters that can be modeled as Kuramotooscillators.

III. APPLICATION TO SMART GRIDS

The smart grid is a large scale power network ofmultiple non-homogeneous nodes with nonlinear cou-pled dynamics. The smart grid possesses a high pene-tration of renewable and distributed sources, includinga communication infrastructure that allows the networkto take smart decisions. Hence, in this work we focusour attention on the frequency synchronization of thenetwork, where our first assumption is that voltagevariations can be neglected, such that we only considerthe active power flow. As a consequence, we can modelthe power network as a set of power generators, DC-sources with inverters, and loads, using the similaritiesbetween the Kuramoto model and the power networkswing equations [7] [8]. Furthermore, the complexityof the problem increases when the connection and dis-connection of nodes are considered. Synchronizationof these isolated nodes is important due to the factthat, when nodes reconnect to the main grid, phase andfrequency need to be synchronized in order to avoidfailures. The power AC network is described by thepurely inductive admittance matrix Y ∈ jRN×N , thenodal voltage magnitude Ei, and the nodal voltage phaseθi ∈ S1. The voltage magnitude is assumed to be

constant, such that aij = |Ei||Ej ||Yij | is the maximumreal power transfer between nodes i and j. Then, weconsider three types of nodes, which are subsets of thevertex set V , such that υ1, υ2, υ3 ⊂ V: i) Synchronousgenerators are described by the swing dynamics

Miθi +Diθi = Pm,i −∑N

j=1aij sin(θi − θj), (10)

where i ∈ υ1, θi is the generator rotor angle, θi isthe frequency or speed of the generator, Pm,i is themechanical power input, and Mi and Di are the inertiaand damping coefficients, respectively. In this work, theinertia term Mi can be omitted in the analysis becauseit only modifies the convergence time to synchronize thestate but not its existence.ii) DC sources connected to a DC/AC inverter equippedwith a conventional droop controller. For each node,there is a nominal power Pd,i > 0, where the droop-slope 1/Di > 0, and the frequency deviation Diθi isproportional to the difference between the output power(∑Nj=1 aij sin(θi − θj)) and the nominal power [11].

Then, the dynamics of the ith inverter, for i ∈ υ2 canbe described by

Diθi = Pd,i −∑N

j=1aij sin(θi − θj), (11)

iii) The load buses consume an amount of power PL,i,with a frequency-dependent term Diθi with Di > 0 thatintroduces the effects of frequency changes in loads, suchthat

Diθi = −PL,i −∑N

j=1aij sin(θi − θj), (12)

for i ∈ υ3. Results can be easily extended for constantpower loads, with Di = 0. However, for the sakeof obtaining a more general result, we only considerfrequency-dependant loads.

Note that Equations (10)-(12) are similar to the classicKuramoto model with multiple time rates., where P =(Pm, Pd,−PL) are equivalent to the natural frequencies.Conditions for frequency synchronization are well de-fined in [8], such that the network synchronizes to afrequency ωsync =

∑Ni=1 Pi∑Ni=1Di

. However, power systemrequires that the frequency in the whole network hasto be equivalent to a desired frequency (e.g., 50 or 60Hz). As a consequence, we can use the proposed controltechnique in (2), such that the network follows a desiredreference, preserving phase cohesiveness.Nevertheless, it is necessary to obtain some design pa-rameters, the maximum injected and minimum absorbedpower, in order to minimize the costs of the storagesystem ensuring synchronization. The next theorem in-troduces the conditions to assure synchronization of apower network including ESS, based on the consensuscontroller proposed in Section II-A.

Theorem 4: Let us consider the power network of Nnodes, with an interconnection topology described bythe weighted undirected graph Gp(V, Ep) with adjacencymatrix Ap. Nodes are described by Equations (10)-(12),where some synchronous generators and DC-sources areconnected to ESS. Assume that the amount of powerinjected/absorbed by the ESS is based on the control

strategy in (2), using the information flow describedby the directed graph Gc(V, Ec). The power networksynchronizes to a desired reference θr if conditions inTheorems 1 and 2 are satisfied for ωi = Pi, and the ESScapabilities umaxi , umini are defined by Theorem 3.Proof: Dorfler et. al. [8] demonstrated that the stabilityand synchronization of Kuramoto oscillator with inertiaterms, similar to the one described in Equation (10), donot depend on the inertia term, and the analysis can beperformed assuming Mi = 0 for i ∈ υ1. Then, if wedefine P = (Pm, Pd,−PL), and we assume that somenodes have control capabilities based on informationtransmitted through the communication infrastructure,the swing equations are equivalent to Equation (7) withinput constraints. Then, synchronization is achieved ifconditions in Theorems 1, 2, and Theorem 3 are satisfied.�

A. Kuramoto Oscillator Under Sampling

An essential design parameter for the smart grid is thecharacteristics of the transmission of information. In thiscontext, we analyze the extended Kuramoto oscillatordynamics for the case when sampled-data measurementsare present. Frequency dynamics can be obtained analyz-ing the frequency differences of the swing equations in(10)-(12), with the control strategy in Section 2 includingthe virtual agent of zero dynamics. As we need toinclude communication limitations, we define θ∗j , whichcorresponds to the frequency data received from the jth

neighbor and it is the sampled representation of θj . Thus,the term KLN+1

B is divided into KD(BN+1c )−KBN+1

c

and we define ¯θ∗ = {θ∗1 , . . . , θ∗N , θr} the vector of datareceived from all nodes. We assume that the sampledsignal is retained with a zero order holder, that provokesthe input of each node to be constant for t ∈ [kδ, (k +1)δ]. Then, the output and input of the whole system aredescribed by y(t) = ¯θ, and v(t) = ¯θ

∗respectively, and

the state representation can be given byd

dtD¯θ = A¯θ +Bv(t) (13)

for A = −(KD(Bc) + LN+1

A (t))

and B = KBN+1c

The following theorem illustrates the sampling periodindependence.

Theorem 5: Synchronization under sampling: As-sume the sampled data synchronization problem in (13),for constant input v∗(t) in the interval t ∈ [δk, δ(k+1)]

and output y(t) = ¯θ. If its continuous-time represen-tation without sampling is strictly passive with storagefunction V (¯θ(t)) = 1

2¯θ(t)>D>QD¯θ(t), for V > 0 and

V (0) = 0, and y(t) = 12 (¯θ(t)+v(t)), then, its sampled-

data representation is also strictly passive independent ofthe sampled period and synchronization is asymptoticallyachieved.

Sketch of the proof:Analysing Equation (13) for the continuous-time casefor directed graphs according to [16], it is easy to verifythatV (

¯θ) = −

1

2¯θ>

(t)(L>(t)QD + DQL(t)

)¯θ(t) ≤ v>(t)y(t).

(14)which assures stability if LN+1

A + BN+1c is connected.

Thus, y(t) = v(t) = ¯θ(t). Then, without loss of gener-ality, we can rewrite the output as y(t) = 1

2 (¯θ(t)+v(t)).

According to [12], a direct input/output link implies thatits sampling data representation is passive and preservesstability.

IV. SIMULATIONS

In order to illustrate the performance of the proposedcontrol methodology, the IEEE 30 bus system withdistributed generation (DG) is studied [17](Figure 1.The system possesses 6 synchronous generators and 21loads.MWe consider some DC-sources with inverters anda maximum capacity of 4 MW. DG sources are placedin nodes 14, 15, 16, 17, 18, 19, 21, 24, 26, 29, 30 and theyhave equal constant D = 0.3MW ·s. First, we study the

AREA 1

AREA 2 AREA 3

Fig. 1: IEEE 30 bus system with eleven distributed generators basedon the results in [17].

case when no control is placed through the network, andarea 1 is isolated. Then, after 10 seconds, it reconnectsback. Figure 2 shows that before the reconnection, both

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

θ i−θj (

rad/

sec)

0 2 4 6 8 10 12 14 16 18 200.95

1

1.05

Time (sec)

Fre

quen

cy (

p.u.

)

Fig. 2: IEEE 30 bus system with eleven distributed generators wherearea 1 is disconnected. Phase differences (top) and frequency (bottom)where the dashed line corresponds to 60 Hz.

subnetworks (one subnetwork formed by Area 1 andother by Areas 2 and 3) synchronize to different fre-quencies. When the reconnection is activated, it inducessome drastic frequency changes and the whole networksynchronizes to ωsync, which is below the desired oneof 60 Hz (1 p.u.). The drastic variations may provokedamages to the generators and undesired oscillations.

On the other hand, when control is included in somenodes, such that the six generator nodes posses controlcapabilities, with ESS designed according to Theorem 3.We assume that the storage device in node 1 and in node22 posses high storing capabilities, of 15 MW. The ESSin nodes 1, 13, 23, 27 are batteries of medium-voltagecapacity (20 KW). Each node with communication capa-bilities transmits its information with a sampling periodof 0.05 sec, according to the typical sampling time ofPMUs. The communication architecture described by agraph of degree 2, i.e., each controller receives informa-tion from two of its neighbors. In Figure 3, frequency

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

θ i−θj (

rad/

sec)

0 2 4 6 8 10 12 14 16 18 200.95

1

1.05

Fre

quen

cy (

p.u.

)

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

0.4

Time (sec)

u i (M

W)

ESS1

ESS2

ESS13

ESS22

ESS23

ESS27Fig. 3: Phase differences (top), control actions where the dashed linesindicate the upper and lower bound of the ESS (bottom), and frequencyvariations where the dashed line represents the reference value of 60Hz (1 p.u.)(middle).tracking is observed even when the physical topology isdisconnected. Hence, thanks to the control input, phasecohesiveness is preserved and the reconnection of area1 does not provoke drastic changes on the frequency ofthe nodes. Moreover, we can observe that the controlinputs (Figure 3 bottom) absorbs or injects power tothe network, and the input of node 27 saturates closeto 4 seconds. Even when the capacity of the flywheelsin nodes 1 and 22 is very high, the full capacity is neverneeded. However, drastic changes in the load or in theadjacent nodes may provoke an increment in the storageutilization.

V. CONCLUSIONS

We have developed a control technique that allowsthe network of Kuramoto oscillators to oscillate to adesired frequency according to a consensus-based con-troller. Then, we have shown that stability and syn-chronization depend on the connectivity of the graphobtained from the two topologies, and the selection of aconstant parameter Ki, for all i ∈ V . Besides, we haveestablished some design parameters when the input isconstrained. All the results have been extended to thesynchronization of power networks, and we have proventhe feasibility of our proposed controller through thesimulation of the IEEE 30 bus system with distributedgeneration.However, other approaches for dealing withsaturation should be considered, and the inclusion ofdelays and the possibility of tracking a time-varyingreference should be also addressed.

VI. ACKNOWLEDGMENTS

The authors would like to thank Florian Dorfler forhis suggestions and assistance in the simulation process.

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