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Synchronization of Nonlinear Oscillators in an LTIElectrical Power Network

Brian B. Johnson, Member, IEEE, Sairaj V. Dhople, Member, IEEE,Abdullah Hamadeh, and Philip T. Krein, Fellow, IEEE

Abstract—Sufficient conditions are derived for the globalasymptotic synchronization of a class of identical nonlinearoscillators coupled through a linear time-invariant network. Inparticular, we focus on systems where oscillators are connectedto a common node through identical branch impedances. Forsuch networks, it is shown that the synchronization conditionis independent of the number of oscillators and the value ofthe load impedance connected to the common node. Theoreticalfindings are then applied towards the analysis and control ofvoltage sourced inverters in a decentralized power system. Theensuing paradigm i) does not necessitate communication betweeninverters, ii) is independent of system load, and iii) facilitates amodular design approach because the synchronization conditionis independent of the number of oscillators. We present bothsimulation and experimental case studies to validate the analyticalresults and demonstrate the proposed application.

Index Terms—inverter control, microgrids, nonlinear oscilla-tors, synchronization.

I. INTRODUCTION

SYNCHRONIZATION of coupled oscillators is relevantto several research areas including neural processes, co-

herency in plasma physics, communications, and electroniccircuits [1]–[7]. This paper presents a sufficient conditionfor global asymptotic synchronization of a class of identicalnonlinear oscillators coupled through a linear time-invariant(LTI) network. In particular, symmetric networks composedof oscillators connected to a common node through iden-tical branch impedances are examined. The synchronizationcondition is independent of: i) the load impedance (i.e., theimpedance connected between the common node and electricalground), and ii) the number of oscillators in the network. Thisresult is used to formulate a control and analysis paradigm fora decentralized power system composed of parallel voltagesource inverters serving a passive electrical load.

B. B. Johnson is with the Power Systems Engineering Center atthe National Renewable Energy Laboratory, Golden, CO, 80401 (e-mail:brian.johnson@NREL.GOV, tel: 303-275-3967), but has written this articleoutside the scope of his employment; S. V. Dhople is with the Departmentof Electrical and Computer Engineering at the University of Minnesota,Minneapolis, MN, email: sdhople@UMN.EDU. A. O. Hamadeh is with theDepartment of Electrical and Computer Engineering at Rutgers University,Piscataway, NJ, email: abdullah.hamadeh@RUTGERS.EDU. P. T. Krein iswith the Department of Electrical and Computer Engineering at the Universityof Illinois, Urbana, IL, email: krein@ILLINOIS.EDU.

B. B. Johnson was supported in part by a National Science FoundationGraduate Research Fellowship and the Grainger Center for Electric Machineryand Electromechanics at the University of Illinois. P. T. Krein was supportedin part by the Global Climate and Energy Project at Stanford University.

Copyright c©2013 IEEE. Personal use of this material is permitted. How-ever, permission to use this material for any other purposes must be obtainedfrom the IEEE by sending an email to pubs-permissions@IEEE.ORG

Relevant to this work is a body of literature that hasexamined synchronization conditions for diffusively coupledoscillators using passivity theory [8]–[12]. For instance, in[13], the notions of passivity and incremental passivity [8]–[12] were used to establish synchronization conditions thatwere applied to the control of inverters as nonlinear oscillatorsin a power system. Passivity-based approaches require theformulation of a storage function, which can be difficultwhen the network contains energy-storage circuit elementssuch as inductors and capacitors. Since power networks arein general composed of a variety of LTI circuit elements(resistors, capacitors, inductors, and transformers), passivity-based approaches are difficult to apply in such systems. Inthis work, we use L2 input-output stability methods, becausethey facilitate analysis in settings where storage functions aredifficult to formulate. Our approach derives from previouswork in [14]–[16] where L2 methods were used to analyzesynchronization in feedback systems. To prove synchroniza-tion, we reformulate the dynamics of the original system in acorresponding differential system based on signal differences.Stability of the differential system implies synchronization inthe original system.

The application focus of this work is a local control ap-proach of power electronic inverters in a self-assembling acmicrogrid. A microgrid is a decentralized electrical powersystem containing generation, storage, and loads that canoperate independent of the bulk power system [17], [18].Microgrids are an enabling technology for decentralized powersystems since they provide a number of advantages including:increasing renewable integration, reducing transmission anddistribution losses, and ensuring a reliable power supply toloads in mission-critical applications. Design objectives of mi-crogrids are generally focused on minimizing communication[19], [20], maintaining stability [21]–[24], and ensuring thatinverters share the load in proportion to their ratings [25], [26].

Inverters perform the key task of power delivery in anac microgrid. With advances in digital control, they can beprogrammed to behave as controllable voltage sources [27].In this work, we propose inverters be controlled to behave asnonlinear oscillators that are designed to synchronize in a mi-crogrid. The control scheme does not require communicationbetween inverters (beyond the coupling inherently introducedby the electrical network), is independent of the systemload (because the synchronization condition is independent ofload impedance), and facilitates a modular design approach(because the synchronization condition is independent of thenumber of oscillators). Also, the inverters share the load power

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demand with no supervisory control effort.The state of the art method for inverter control in microgrids

is droop control. This method requires no communicationand is based on modulating the inverter output such that thefrequency and voltage amplitude are inversely proportional tothe real and reactive power output, respectively [20], [28], [29].Recently, synchronization of droop-controlled inverters hasbeen analyzed with equivalent Kuramoto-oscillator models,and sufficient conditions for convergence and stability havebeen obtained [30]–[32]. The oscillator-based method wepropose differs from droop control in several respects. In par-ticular, the proposed approach: i) does not require computationof the real and reactive power output, ii) demonstrates minimaldeviations in system frequency from the rated value regardlessof fluctuations in the load impedance, and iii) does not requirean explicit frequency and amplitude command for the inverterac output.

It is foreseen that the analytical results in this work willprovide broad theoretical utility while outlining a compellingapplication to the control of inverters in microgrids. To sum-marize, the contributions of this work are as follows:

1) A sufficient global asymptotic synchronization condi-tion is derived for a class of identical nonlinear os-cillators connected to a common node through branchimpedances.

2) It is shown that the synchronization condition is in-dependent of the number of oscillators and the loadimpedance.

3) These results are applied towards the coordination ofinverters in a single-phase microgrid to achieve a controland design paradigm that is robust (independent of load)and modular (independent of number of inverters).

The remainder of this paper is organized as follows: Notationand background material are presented in Section II. Wedescribe the network topology of interest in Section III, andderive sufficient conditions for global asymptotic synchroniza-tion of the nonlinear oscillators in Section IV. In SectionV, we formulate an oscillator model for inverter control andpresent simulations and experimental case studies. Concludingremarks are given in Section VI.

II. PRELIMINARIES

For the N -tuple (u1, . . . , uN ), denote u = [u1, . . . , uN ]T

to be the corresponding column vector, where T indicatestransposition. The N -dimensional column vectors of all onesand all zeros are denoted by 1 and 0, respectively.

The Laplace transform of the continuous-time function f(t)is denoted by f(s), where s = ρ + jω ∈ C, and j =

√−1.

Transfer functions are denoted by lower-case z(s), and transfermatrices are denoted by upper-case Z(s).

The Euclidean norm of a real or complex vector, u, isdenoted by ‖u‖2 and is defined as

‖u‖2 =√u∗u, (1)

where ∗ indicates the conjugate transpose. For somecontinuous-time function u(t), u : [0,∞)→ RN , the L2 norm

of u is defined as

‖u‖L2=

√√√√√ ∞∫0

u (t)Tu (t) dt, (2)

and the space of piecewise-continuous and square-integrablefunctions where ‖u‖L2

<∞ is denoted by L2 [33]. A causalsystem, H , with input u and output y, is said to be finite-gainL2 stable if there exist finite, non-negative constants, γ andη, such that

‖y‖L2= ‖H (u)‖L2

≤ γ ‖u‖L2+ η, ∀u ∈ L2. (3)

The smallest value of γ for which there exists a η such that (3)is satisfied is called the L2 gain of the system and is denotedby γ (H). If H is a linear system and can be represented bythe transfer function H (s), it can be shown that the L2 gainof H is equal to its H-infinity norm, denoted by ‖H‖∞, anddefined as

γ (H) = ‖H‖∞ = supω∈R

‖H (jω)u (jω)‖2‖u (jω)‖2

, (4)

where ‖u (jω)‖2 = 1, provided that all poles of H(s) havestrictly negative real parts [34]. Note that if H (s) is a single-input single-output transfer function then γ (H) = ‖H‖∞ =supω∈R‖H (jω)‖2.

The electrical system of oscillators which interface to acommon node corresponds to a network with all-to-all cou-pling. The Laplacian matrix of such a network is denoted asΓ ∈ RN×N , and given by

Γ := NIN − 11T =

N − 1 −1 · · · −1−1 N − 1 · · · −1

......

. . ....

−1 −1 · · · N − 1

. (5)

This particular Laplacian has the following properties: i)rank(Γ) = N − 1, ii) the eigenvalues of Γ are denoted byλ1 < λ2 = · · · = λN , where λ1 = 0 and λj = N forj = 2, . . . N , iii) Γ is symmetric with row and column sumsequal to zero such that Γ1 = ΓT1 = 0, iv) the eigenvector q1(corresponding to λ1 = 0) is given by q1 = 1√

N1, and v) Γ

can diagonalized as QΛQT, where it follows that Q−1 = QT

because Γ = ΓT. See [15], [35], [36] for proofs and additionaldiscussion.

A useful construct that will be employed to compare indi-vidual oscillator outputs with the average of all N oscillatoroutputs is the projector matrix, Π, defined as [9], [11], [15]

Π = IN −1

N11T. (6)

For some vector u ∈ RN , we will denote u = Πu, and refer tou as the corresponding differential vector (in previous work,see, e.g., [9], [11], [14]–[16], the quantity u is referred to asan incremental quantity). A causal system, H , with input uand output y, is said to be differentially finite-gain L2 stableif there exist finite, non-negative constants, γ and η, such that

‖y‖L2≤ γ ‖u‖L2

+ η, ∀u ∈ L2, (7)

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Figure 1: The oscillator model in this work is composed of alinear subsystem, zosc(s), and a nonlinear voltage-dependentcurrent source, g(ν).

where y = Πy. The smallest value of γ for which there existsa η such that (7) is satisfied is called the differential L2 gainof the system and is denoted by γ (H).

III. SYSTEM OF COUPLED OSCILLATORS

In this section, a system of N nonlinear coupled oscillatorsis introduced. We consider a network topology where alloscillators are connected to a common node through identi-cal impedances. A corresponding differential system is thendeveloped to facilitate synchronization analysis.

Our interest in this particular topology stems from the factthat inverters in a microgrid are often connected in parallelto serve a load [20], [37]. In the forthcoming section, wewill show that the synchronization condition for this networkis independent of the number of oscillators and the loadimpedance. This is a useful result, because it implies thesystem can be designed without a priori information of theload parameters and number of inverters.

A. System Description

As shown in Fig. 1, the electrical oscillator under consid-eration has: i) a linear subsystem composed of passive circuitelements with impedance zosc(s), and ii) a nonlinear voltage-dependent current source g(·). We will require that g(·) becontinuous and differentiable, and additionally impose

σ := supν∈R

∣∣∣∣ ddνg(ν)

∣∣∣∣ <∞. (8)

Figure 2 depicts an electrical network of N such oscillatorsconnected to a common (load) node through identical branchimpedances, znet(s) (which may contain any combination oflinear circuit elements). The N oscillators deliver power to apassive LTI load, zload(s), which is connected at the commonnode. The coupling between oscillators is captured by

i(s) = Y (s)v(s), (9)

where i(s) = [i1(s), . . . , iN (s)]T is the vector of oscillatoroutput currents, v(s) = [v1(s), . . . , vN (s)]T is the vector ofoscillator terminal voltages, and Y (s) is the network admit-tance matrix.

A closed-form expression for Y (s) will now be derived.Towards this end, notice from Fig. 2 that the jth oscillatoroutput current is given by

ij (s) =1

znet (s)(vj (s)− vload (s)) , (10)

Electrical Network:Oscillator #1

Oscillator #2

Oscillator #N

Figure 2: N parallel oscillators interconnected through an LTIelectrical network containing a passive load.

where the load voltage, vload (s), can be expressed as

vload (s) = zload (s)

N∑k=1

ik (s) . (11)

Substituting (11) in (10) yields

ij (s) =1

znet (s)

(vj (s)− zload (s)

N∑k=1

ik (s)

), (12)

from which vj (s) can be isolated to obtain

vj (s) = znet(s)ij (s) + zload (s)N∑k=1

ik (s) . (13)

Collecting all terminal voltages,

v (s) =(znet (s) IN + zload (s)11T) i (s) . (14)

Comparing (14) with (9) indicates

Y −1 (s) = znet (s) IN + zload (s)11T. (15)

The above expression can be inverted to obtain

Y (s) = α(s)IN + β(s)Γ, (16)

where α(s), β(s) ∈ C are given byα(s) = (znet(s) +Nzload(s))

−1=: z−1eq (s),

β(s) = zload(s) (znet(s)zeq(s))−1.

(17)

See Appendix A for a proof of the above result. Using theexpressions for Y (s) in (16)–(17), it is possible to redraw themicrogrid network in Fig. 2 as another equivalent networkcontaining admittances α(s) and β(s).

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Figure 3: Representation of coupled oscillator system. Thelinear and nonlinear portions of the system are compartmen-talized in F (·, ·) and g(·), respectively.

We now seek a representation of the system in Fig. 2, wherethe linear and nonlinear portions of the system are clearlydifferentiated. Towards this end, first note that the terminalvoltage of the jth oscillator, vj(s), can be expressed as

vj(s) = zosc(s) (isrcj(s)− ij(s)) . (18)

Writing all vj’s in matrix form yields

v(s) = Zosc(s) (isrc(s)− i(s))= Zosc(s)isrc(s)− Zosc (s)Y (s) v (s) , (19)

where Zosc(s) = zosc(s) · IN ∈ CN×N , isrc(s) =[isrc1(s), . . . , isrcN (s)]T, and in the second line of (19), wehave substituted i(s) = Y (s)v(s) from (9). We can isolatev(s) from (19) as follows:

v(s) = (IN + Zosc (s)Y (s))−1Zosc (s) isrc (s)

= F (Zosc (s) , Y (s)) isrc (s) , (20)

where F : CN×N ×CN×N → CN×N is the linear fractionaltransformation, and represents Zosc (s) in negative feedbackwith Y (s) [38]. In general, for some A,B of appropriatedimension and domain, the linear fractional transformation isdefined as

F (A,B) := (IN +AB)−1A. (21)

Using (20), the system of coupled oscillators admits thecompact block-diagram representation in Fig. 3. The linear andnonlinear portions of the system are clearly compartmental-ized by F (Zosc (s) , Y (s)) and g(v) := [g(v1), . . . , g(vN )]T,respectively.

B. Corresponding Differential System Description

Global asymptotic synchronization of the coupled oscillatorsystem in Section III-A corresponds to the condition

limt→∞

vj(t)− vk(t) = 0 ∀j, k = 1, . . . , N. (22)

For ease of analysis, we will find it useful to transform to acoordinate system based on signal differences. Towards thisend, we employ the projector matrix in (6), noting that [14],[15]

v(t)Tv(t) = (Πv(t))T

(Πv(t)) =1

2N

N∑j=1

N∑k=1

(vj(t)− vk(t))2.

(23)Therefore, it is evident that the synchronization condition in(22) is equivalent to requiring v(t) = Πv(t) → 0 as t →∞. Henceforth, we will refer to the system where all vectorsare transformed by the projector matrix as the correspondingdifferential system.

Figure 4: Block-diagram representation of the correspondingdifferential system. The linear and nonlinear portions of thesystem are compartmentalized in F (·, ·) and g, respectively.

We will now use the dynamics of the original system toconstruct the corresponding differential system, the stability ofwhich will imply synchronization in the sense of (22). Towardsthis end, the differential terminal-voltage vector, v(s), can beexpressed as

v(s) = Πv(s) = Π (Zosc(s) (isrc (s)− i (s)))

= Zosc(s) (Πisrc(s)−ΠY (s)v(s))

= Zosc(s)(isrc(s)− Y (s)v(s)

), (24)

where in the first line, we have substituted for v(s) from (19);in the second line, we have used the relation i(s) = Y (s)v(s)from (9) and the fact that ΠZosc(s) = Πzosc (s) IN =zosc (s) INΠ = Zosc(s)Π; and in the last line, we have used theproperty that the projector and admittance matrices commute,i.e., ΠY (s) = Y (s)Π (this follows straightforwardly from thefact that ΓΠ = ΠΓ). We can now isolate v(s) in (24) asfollows:

v(s) = (IN + Zosc(s)Y (s))−1Zosc(s)isrc(s)

= F (Zosc(s), Y (s)) isrc(s). (25)

Notice the similarity between (25) and (20); i.e., the linearfractional transformation also maps isrc(s) to v(s).

Finally, we can define a map g that captures the impact ofg(v) in the corresponding differential system as follows:

g : v → −isrc. (26)

We now have a complete description of the correspondingdifferential system. In particular, this system admits the block-diagram representation in Fig. 4, where, as in Fig. 3, the linearand nonlinear subsystems are compartmentalized using F (·, ·)and g, respectively.

IV. GLOBAL ASYMPTOTIC SYNCHRONIZATION

In this section, we derive a sufficient condition that ensuresglobal asymptotic synchronization in the sense of (22) for thesystem of oscillators described in Section III-A.

First, we present a lemma that will establish an upper boundon the differential L2 gain of g.

Lemma 1. The differential L2 gain of g is finite and upperbounded by σ:

γ (g) ≤ σ = supν∈R

∣∣∣∣ d

dνg(ν)

∣∣∣∣ <∞. (27)

Proof: By definition of σ, for any pair of terminal voltagesvj and vk, and the corresponding currents isrcj and isrck,

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where j, k ∈ 1, . . . , N, the mean-value theorem [34] canbe applied to give

σ ≥ |isrcj (t)− isrck (t)||vj (t)− vk (t)|

=⇒ σ2 (vj (t)− vk (t))2 ≥ (isrcj (t)− isrck (t))

2. (28)

Summing over all indices, j, k ∈ 1, . . . , N, we arrive at

σ2N∑j=1

N∑k=1

(vj (t)− vk (t))2 ≥

N∑j=1

N∑k=1

(isrcj (t)− isrck (t))2,

(29)which can be rearranged and simplified as

σ ≥

√√√√∑Nj=1

∑Nk=1 (isrcj (t)− isrck (t))

2∑Nj=1

∑Nk=1 (vj (t)− vk (t))

2. (30)

Since (30) holds for any set of terminal voltages, we obtain

σ ≥ supv∈RN

√√√√ 12N

∑Nj=1

∑Nk=1 (isrcj (t)− isrck (t))

2

12N

∑Nj=1

∑Nk=1 (vj (t)− vk (t))

2, (31)

which can be rewritten compactly using the projector-matrixnotation in (23) as follows:

σ ≥ supv∈RN

√isrc(t)Tisrc(t)

v(t)Tv(t). (32)

By definition of the differential L2 gain, we have

γ (g) = supv∈RN

∥∥∥isrc

∥∥∥L2

‖v‖L2

= supv∈RN

√∫∞0isrc(t)Tisrc(t) dt√∫∞

0v(t)Tv(t) dt

. (33)

Applying (32) in the definition above, we attain

γ (g) ≤ supv∈RN

√σ2∫∞0v(t)Tv(t) dt∫∞

0v(t)Tv(t) dt

= σ <∞, (34)

which completes the proof.We now prove the main result of this work: a sufficient

condition for global asymptotic synchronization in the networkof oscillators described in Section III-A.

Theorem 1. The network of N oscillators coupled through(9) with the admittance matrix as defined in (16)–(17), syn-chronizes in the sense of (22), if

supω∈R

∥∥∥∥ znet(jω)zosc (jω)

znet(jω) + zosc (jω)

∥∥∥∥2

σ < 1. (35)

Proof: Consider the block-diagram of the differentialsystem in Fig. 4. Denoting the differential L2 gain of the linearfractional transformation by γ (F (Zosc(s), Y (s))), we have

‖v‖L2≤ γ (F (Zosc(s), Y (s)))

∥∥∥isrc

∥∥∥L2

+ η, (36)

for some non-negative η. From Lemma 1, we also have∥∥∥isrc

∥∥∥L2

≤ σ ‖v‖L2. (37)

Combining (36) and (37), we arrive at

‖v‖L2≤ γ (F (Zosc(s), Y (s)))σ ‖v‖L2

+ η. (38)

Let us assume that

γ (F (Zosc(s), Y (s)))σ < 1. (39)

Isolating ‖v‖L2from (38), we can write

‖v‖L2≤ η

1− γ (F (Zosc(s), C(s)))σ, (40)

which implies that v ∈ L2. It follows from Barbalat’s lemma[14]–[16], [34] that

limt→∞

v(t) = 0 =⇒ limt→∞

vj(t)− vk(t) = 0 ∀j, k = 1, . . . , N.

(41)That is, if the system of oscillators satisfies the condition in(39), global asymptotic synchronization can be guaranteed.

We will now derive the result in (35) by showingγ (F (Zosc(s), Y (s))) =

∥∥∥ znet(jω)zosc(jω)znet(jω)+zosc(jω)

∥∥∥∞

. From the defini-tion of the linear fractional transformation in (21) and thegeneral form of the admittance matrix in (16), note that

F (Zosc(s), Y (s)) = (IN + Zosc(s)Y (s))−1Zosc(s)

= (IN + Zosc(s) (α(s)IN + β(s)Γ))−1Zosc(s)

=

(IN +

zosc(s)

1 + α(s)zosc(s)β(s)Γ

)−1zosc(s)IN

1 + α(s)zosc(s)

= F (ζ(s)IN , β(s)Γ) , (42)

where we have defined

ζ(s) :=zosc (s)

1 + α(s)zosc(s), (43)

with α(s) and β(s) given in (17). Now, by definition of thedifferential L2 gain of the linear fractional transformation, wecan express

γ (F (Zosc(s), Y (s))) = γ (F (ζ(s)IN , β(s)Γ))

= supω∈R

∥∥∥F (ζ(s)IN , β(s)Γ) isrc (jω)∥∥∥2∥∥∥isrc(jω)

∥∥∥2

= supω∈R

∥∥∥(IN + ζ(jω)β(jω)Γ)−1ζ(jω)isrc (jω)

∥∥∥2∥∥∥isrc(jω)

∥∥∥2

= supω∈R

∥∥∥Q (IN + ζ(jω)β(jω)Λ)−1ζ(jω)QTisrc(jω)

∥∥∥2∥∥∥QTisrc(jω)

∥∥∥2

, (44)

where, we have diagonalized Γ = QΛQT and recognizedthat QQT = IN . We will now make two key observationsto simplify (44):

i) The first column of Q is given by q1 = 1√N1. Fur-

thermore, 1TΠ = 0T. Therefore, the vector QTisrc(s) =QTΠisrc(s) is given by

QTisrc(s) = QTΠisrc(s) = [0, D(s)]T, (45)

where D(s) ∈ CN−1×1 is made up of the non-zero elementsof QTΠisrc(s).

ii) Denote the diagonal matrix with entries made up of thenon-zero eigenvalues of Γ by ΛN−1. By definition of Γ inSection II, we see that ΛN−1 = N · IN−1 ∈ RN−1×N−1.

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(a) (b)

Figure 5: The functions (a)f(v) and (b)g(v) illustrated forthe Van der Pol (dashed lines) and dead-zone (solid lines)

oscillators. For the dead-zone oscillator, supv∈R

∣∣∣∣ ddvg(v)

∣∣∣∣ = σ.

Using the two observations highlighted above, we can nowsimplify (44) as follows:

γ (F (ζ(s)IN , β(s)Γ))

= supω∈R

∥∥∥(IN−1 + ζ(jω)β(jω)ΛN−1)−1ζ(jω)D(jω)

∥∥∥2

‖D(jω)‖2

= supω∈R

(1

DT(jω)D(jω)

ζ2(jω)DT(jω)D(jω)

(1 + ζ(jω)β(jω)N)2

) 12

= supω∈R

(1 + ζ(jω)β(jω)N)−1ζ(jω)

= ‖F (ζ(s), β (s)N)‖∞ . (46)

Finally, for the α(s) and β(s) in (17), we can simplifyF (ζ(s), β (s)N) as follows:

F (ζ(s), β (s)N) = (1 + ζ(s)β(s)N)−1ζ(s)

=zosczeq (zosc + zeq)

−1

1 + zosczeq (zosc + zeq) zloadz−1net z

−1eq N

=zosc

z−1eq (zosc + zeq) + zoscz−1eq zloadz

−1net N

=zosc

1 + zoscz−1eq(1 + zloadz

−1net N

)=

zosc

1 + zoscz−1eq(zeqz

−1net) =

zosc(s)znet(s)

znet(s) + zosc(s), (47)

which completes the proof.The condition for synchronization in (35) is independent of

the number of oscillators, N , and the load impedance, zload(s).It depends only on the impedance of the parallel combinationof zosc(s) and znet(s).

In Appendix C, we discuss how the results presented abovecan be extended to the case where the oscillators are nothomogeneous, and the network branch impedances are notidentical.

V. CASE STUDIES

The proposed microgrid control is grounded on program-ming inverters to emulate nonlinear oscillators. Towards thatend, we first present the particular oscillator model which willform the basis of the inverter control. Next, we describe the pa-rameter selection approach to ensure the inverters synchronizewhile ensuring the load voltage and system frequency meet

-1 0

0

-1

1

2

3

-2

-31

(a)

-1 0

0

-1

1

2

3

-2

-31

(b)

Figure 6: Phase plot of steady-state limit cycles in the (a)dead-zone and (b) Van der Pol oscillators for varying ε =√

LC

(σ − 1

R

).

performance objectives. We also describe how the controllercan be implemented on a digital platform. Finally, simulationand experimental case studies that demonstrate the analyticalresults as applied to inverters in a microgrid are presented.

A. Nonlinear Oscillator Description

In this work, we utilize a dead-zone oscillator, in which thenonlinear voltage-dependent current source is given by

g(v) = f(v)− σv, (48)

where f(·) is a continuous, differentiable dead-zone functionwith slope 2σ, and f(v) ≡ 0 for v ∈ (−ϕ,+ϕ). Furthermore,the linear subsystem is a parallel RLC circuit:

zosc(s) = R || sL || (sC)−1. (49)

The functions f(v) and g(v) are illustrated in Fig. 5.1 Theterminal voltage of the dead-zone oscillator satisfies

LCd2v

dt2+ L

(df(v)

dv+

1

R− σ

)dv

dt+ v = 0. (50)

The existence of a stable and unique limit cycle can bedetermined with the aid of Liénard’s Theorem, which is statedbelow.

Theorem 2. (Liénard’s Theorem [4]) Consider the system

v + r (v) v +m (v) = 0, (51)

where v : [0,∞) → R and r (v) ,m (v) : R → R aredifferentiable, even and odd functions, respectively. Define

R (v) :=

∫ v

0

r (τ) dτ. (52)

The system in (51) has a unique and stable limit cycle if:i) m (v) > 0 ∀v > 0, ii) R (v) has one positive zero for

1The proposed dead-zone oscillator is very similar to the well-known Vander Pol oscillator. To facilitate comparison, the functions f(v) and g(v) forthe Van-der-Pol oscillator are superimposed in Fig. 5.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: REGULAR PAPERS (ACCEPTED, 2013) 7

−+

netz

)Cv(g

−

+

Virtual Oscillator

to rest of

system

Digital Control

m

R L C

PWM

Cv i

Figure 7: Controller implementation that ensures the H-bridgeinverter emulates the behavior of the nonlinear dead-zoneoscillator.

some v = p, iii) R (v) < 0 when 0 < v < p, and iv) R (v)monotonically increases for v > p and lim

v→∞R (v) =∞.

We can rewrite (50) by expressing the derivatives of v withrespect to ξ := t/

√LC to yield

v +

√L

C

(df (v)

dv+

1

R− σ

)v + v = 0, (53)

which is of the form in (51), withm (v) = v,

r (v) =√

LC

(df(v)dv + 1

R − σ).

(54)

For the case σ > 1/R, it is easy to see that m(v), r(v), andR(v) satisfy the conditions in Liénard’s theorem, implying thatthe dead-zone oscillator has a stable and unique limit cycle.The steady-state limit cycles of the dead-zone oscillator fordifferent values of ε =

√LC

(σ − 1

R

)are plotted in Fig. 6(a).

For comparison, the limit cycles of the Van der Pol oscillatorfor the same set of parameters are shown in Fig. 6(b). Whenε 1, it can be shown that the steady-state oscillation willhave a frequency approximately equal to 1/

√LC [34], and as

shown in Fig. 6(a), the limit cycle is approximately a circlein the current-voltage space.

B. Parameter Selection and Controller Implementation

In all the case studies, we consider a network with thetopology in Fig. 2. The network branch impedance is givenby znet (s) = sLnet + Rnet, where Lnet and Rnet equalthe series combination of the line and inverter-output-filterinductance and resistance, respectively (the inverter output-filter inductance reduces harmonics which arise due to switch-ing [27]). Finally, we assume the load is resistive such thatzload (s) = Rload (note that our result applies in general toany passive LTI load). For this system, the linear fractionaltransformation is given by

F(zosc(s), z

−1net (s)

)=

a2s2 + a1s

b3s3 + b2s2 + b1s+ b0, (55)

where a2 = Lnet, a1 = Rnet, b3 = LnetC, b2 = (Lnet/R) +RnetC, b1 = (Lnet/L) + (Rnet/R) + 1, and b0 = Rnet/L.

The design objective is to guarantee that the inverterssynchronize their voltage outputs, and oscillate at the desiredfrequency. Additionally, in steady-state we will require vloadto stay within ±5% of the rated voltage across the entire loadrange (no-load to maximum rated load). For a given filterimpedance, znet(s), the above design objective can be satisfiedby proper selection of the oscillator linear- and nonlinearsubsystem parameters including: R, L, C, σ, and ϕ.

To ensure oscillations at the rated system frequency, ωrated,the values of R, L, and C must be selected such thatR > 1/σ, LC = 1/ω2

rated. Further, we must ensure that

ε =√

LC

(σ − 1

R

)is minimized to guarantee that the terminal

inverter voltages are sinusoidal, and σ is picked so that∥∥F (zosc(jω), z−1net(jω)∥∥∞ σ < 1 to guarantee synchroniza-

tion. The value of ϕ can be tuned with an open-circuit testto ensure that the inverter output voltage is no more than thepeak allowed load voltage.

Figure 7 illustrates how the oscillator-based controller isimplemented for a single-phase H-bridge inverter. As shown,the terminal current of the inverter is measured, and extractedfrom the virtual oscillator. The modulation signal, m, isthe scaled oscillator voltage. The inverter switching signalsare generated by comparing the modulation signal with atriangle carrier waveform [27]. With the proposed method,the inverter emulates the dynamics of the nonlinear dead-zoneoscillator. Discretization of the virtual oscillator differentialequations is straightforward, and the proposed controller canbe implemented on a standard microcontroller.

C. Simulation and Experimental Results

We now present simulation and experimental results tovalidate the analytical methods in a microgrid application. Inparticular, we demonstrate that a system of inverters controlledas deadzone oscillators satisfying (35) synchronize and deliverpower to a load.

-200-100 200100 3000

2

1.5

1

0.5

50

0

0

-0.5

-1

-1.5

-2

-50-300

Figure 8: Evolution of oscillator state variables during startupin the presence of a load. Waveforms for only 10 inverters outof 100 simulated are shown for clarity.

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0

2

4

6

-20-40

-400-200

50 100 150 200

Load step

250 300 350 400

40

20

200

400

0

0

Figure 9: Inverter output currents, voltages, andvoltage synchronization error in the case when∥∥F (zosc(jω), z−1net (jω

)∥∥∞ σ < 1.

0

0.1

0.2

0.3

-0.5-1

-1.5

-400-200

50 100 150 200 250 300 350 400

11.5

0.5

200

400

0

0

Figure 10: Inverter output currents, voltages, and voltagesynchronization error when

∥∥F (zosc(jω), z−1net (jω)∥∥∞ σ > 1

and synchronization is not guaranteed.

In case studies I and II, a microgrid consisting of 100 par-allel inverters which are each rated for 10 kW was simulated.The RMS voltage and frequency ratings of the system are220 V and 60 Hz, respectively, and the maximum load poweris 1 MW. In case study III, we provide experimental resultsfor a laboratory prototype which consists of three parallelinverters. The system parameters used in the case studies aresummarized in Table I in Appendix B.

Case Study I (Simulation): Substituting the parametervalues from Table I into (55), it can be shown that∥∥F (zosc(jω), z−1net (jω

)∥∥∞ σ = 0.77 < 1, which guarantees

synchronization. At t = 0, all currents are zero and theoscillator capacitor voltages are randomly selected between±10 V. Initially, the system contains no load. After successfulsynchronization, the load is abruptly added at t = 300 ms. Asshown in Fig. 9, the voltage stays within ±5% of the ratedvalue in steady state.

control

control

control

100 V

100 V

100 V

Figure 11: Schematic of the electrical circuit in the experi-mental setup.

Figure 12: Oscilloscope screenshot for measured inverter out-put currents and load voltage.

A second simulation was conducted to demonstrate synchro-nization in the presence of the load (in other words, the loadis connected at t = 0 s). Given the same initial conditionsas above, Fig. 8 illustrates the trajectories of the oscillatorstate variables (only 10 out of 100 waveforms are shown forclarity). The inductor current within the oscillator RLC circuitis denoted as iL. As shown in Fig. 8, the state-variables reacha stable limit cycle.

Case Study II (Simulation): All parameters, except Rnet,were reused. The value of Rnet was reduced such that∥∥F (zosc(jω), z−1net (jω

)∥∥∞ σ = 2.78 ≮ 1, and synchronization

is not guaranteed. As illustrated in Fig. 10, the inverters donot synchronize.

Case Study III (Experimental): We have built a hardwareprototype comprising three parallel H-bridge inverters and aresistive load. A schematic of the experimental hardware setupis given in Fig. 11. The switches in the schematic are N-channel power MOSFETs. Each inverter is rated to deliver50 W and supplied by a 100 V dc voltage source at the input.The controllers in Fig. 11 regulate the switching action suchthat each inverter behaves like a dead-zone oscillator. Cor-responding parameters in Table I were selected such that the

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system oscillates at 60 Hz while maintaining a 60 V±5% RMSvoltage for all load conditions. Furthermore, synchronizationis guaranteed since

∥∥F (zosc(jω), z−1net (jω)∥∥∞ σ = 0.93 < 1.

Figure 12 shows the output currents and load voltages duringstartup with a 72 W load. The oscillator capacitor voltageswere initialized to v (0) = [5 V, 4 V, 3 V]T within each con-troller to demonstrate synchronization in spite of non-identicalintitial conditions.

VI. CONCLUDING REMARKS AND DIRECTIONS FORFUTURE WORK

A synchronization condition for nonlinear oscillators cou-pled through a symmetric LTI network was derived. Thecondition was shown to be independent of the number ofoscillators and the load parameters. We also proposed that par-allel inverters in a microgrid be controlled to act as dead-zoneoscillators. The resulting microgrid design is modular and doesnot require communication between inverters. Simulation andexperimental results were used to substantiate the analyticalresults and illustrate the merit of the proposed application.

An important direction for future work is to extend themethod presented to address synchronization in three-phaseinverters with constant-power loads. Additionally, synchro-nization in other network topologies can be investigated withthe general approach that we have outlined in this work.Finally, the power quality delivered to the load needs to beinvestigated by studying the impact of oscillator parameterson the harmonic content in the inverter output waveforms.

ACKNOWLEDGEMENT

The authors would like to thank Professor Guy-Bart Stanfor many helpful discussions.

APPENDIX

A. Derivation of Y (s) in (16)

Diagonalizing 11T = PΞP T, we can express (15) asfollows:

Y −1 (s) = znet (s) IN + zload (s)PΞP T

= znet(s)P

(IN +

zload(s)

znet(s)Ξ

)P T. (56)

It is easy to show that Ξ = diag0, . . . , 0, N ∈ RN×N .Substituting this in (56), we then obtain

Y −1(s) = znet(s)Pdiag

1, . . . , 1, 1 +zload(s)

znet(s)N

P T.

(57)Inverting (57), we get

Y (s) =1

znet(s)Pdiag

1, . . . , 1,1

1 + zload(s)znet(s)

N

P T. (58)

From the definition of zeq(s) in (17), we can write

Y (s) =1

znet(s)Pdiag

1, . . . , 1, 1− zload(s)

znet(s)N

P T

=1

znet(s)P

(IN −

zload(s)

zeq(s)Ξ

)P T. (59)

This can be simplified as follows:

Y (s) =1

znet(s)zeq(s)P (zeq(s)IN − zload(s)Ξ)P T

=1

znet(s)zeq(s)

((znet(s) +Nzload(s)) IN − zload(s)11T)

=1

znet(s)zeq(s)

(znet(s)IN + zload(s)

(NIN − 11T))

=1

znet(s)zeq(s)(znet(s)IN + zload(s)Γ) , (60)

where in the last line, we have used the definition of Γ from(5). Notice that (60) is in the same form as (16), with α(s)and β(s) given in (17).

B. Parameters for Simulations and Experiments

The inverter and electrical network parameters used in thesimulation and experimental case studies are given below.

Table I: System parameters used in the case studies.

Case Study I, II Case Study IIISimulation Experiment

N 100 3R 8.66 Ω 95.46 ΩL 433.2µH 4.77 mHC 16.2 mF 1.47 mFσ 1.15 S 104.8 mSϕ 146.1 V 39.8 Vε 0.17 0.17

Rnet 0.1 Ω, 0.02 Ω 1 ΩLnet 500µH 6 mHRload 91.96 mΩ 50 Ω

C. Heterogeneous Oscillators and Nonidentical BranchImpedances

For real-world microgrid applications, it is unlikely thatthe branch impedances of the network are identical, and theoscillators (inverters) are homogeneous. However, we canincorporate these sources of uncertainty in the synchronizationcondition straightforwardly.

First, we attempt to give conditions under which smalldifferences among oscillators leads to small synchronizationerrors. Towards this end, replace the matrix Zosc(s) withZosc(s) = Zosc(s) + ∆(s) = zosc(s)I + ∆(s), where zosc(s)represents the nominal oscillator impedance and ∆(s) is adiagonal matrix that captures deviations of each actual oscil-lator impedance from the nominal value. With this setup, (25)is modified to

v(s) = F(Zosc(s), Y (s))isrc(s) + e(s), (61)

where e(s) = Π∆(s)(isrc(s) − Y (s)v(s)). Combining (61)with the nonlinear subsystem g(·), we find that if the condition∥∥F(Zosc(s), Y (s))

∥∥2σ < 1 (which follows from Theorem 1),

is satisfied, then the feedback combination of (61) with isrc =−g(v) will synchronize when e(s) ≡ 0. Furthermore, takingthe L∞ norm of both sides of (61)

‖v‖L∞ ≤ ‖F(Zosc, Y )‖L1‖isrc‖L∞ + ‖e‖L∞ + η, (62)

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: REGULAR PAPERS (ACCEPTED, 2013) 10

for some constant η that depends on initial conditions. In ad-dition, it is straightforward to see that the nonlinear subsystemis such that ‖isrc‖L∞ ≤ σ‖v‖L∞ . Using the same small-gainargument employed in the case of identical oscillators, thesynchronization error in v will be in L∞ if network solutionsare bounded such that e ∈ L∞ and if ‖F(Zosc, Y )‖L1

σ < 1since this would guarantee

‖v‖L∞ ≤‖e‖L∞ + η

1− ‖F(Zosc, Y )‖L1σ<∞. (63)

In other words, for a network of inverters with differentzosc(s), the maximal synchronization error over all timewill be upper bounded so long as the small gain condi-tion ‖F(Zosc, Y )‖L1σ < 1 holds, and all network signalsare bounded. Further, this bound will be governed by themaximum value of e over time, according to the relation(63). Since e depends directly on ∆(s) (which quantifies thedifferences between network nodes), smaller ∆(s) will reducethe maximal asymptotic synchronization error in v to theextent that if ∆(s) is equal to the zero matrix, this error willalso reduce to zero, retrieving the original result of Theorem 1.

Now consider that the branch impedances are not identicallyequal to znet(s) ∈ C, but for the system of N inverters, theyare given by the vector znet(s) = [znet1(s), . . . , znetN (s)]T ∈CN . For the m-th branch impedance, we will denoteznetm(s) = κmznet(s), where κm ∈ R, and znet(s) is somenominal branch impedance. We will also find it useful to definethe vector κ = [κ−11 , . . . , κ−1N ]T ∈ RN . In this case, withreference to Fig. 7, we would design the controller for them-th inverter to extract the current κmim. With this setup, itis straightforward to show that the admittance matrix is givenby the following:

Y (s) = α(s)IN + β(s)Γ, (64)

where α(s) = (znet(s) +Nξ)−1

=: z−1eq (s) and β(s) =

ξ (znet(s)zeq(s))−1, with ξ = zload(s)(κTi(s))/(1Ti(s)).

Using the admittance matrix in (64) and applying the sameanalysis in Theorem 1, we get the synchronization condition:

supω∈R

∥∥∥∥ znet(jω)zosc (jω)

znet(jω) + zosc (jω)

∥∥∥∥2

σ < 1. (65)

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Brian B. Johnson (S’08, M’13) received the B.S. degree in physics fromTexas State University, San Marcos, in 2008. He received the M.S. and Ph.D.degrees in electrical and computer engineering from the University of Illinoisat Urbana- Champaign, Urbana, in 2010 and 2013, respectively. He is currentlyan Electrical Engineer with the National Renewable Energy Laboratory inGolden, CO.

He was awarded a National Science Foundation Graduate Research Fel-lowship in 2010. His research interests are in power electronics, distributedgeneration, and renewable energy systems, and nonlinear controls.

Sairaj V. Dhople (S’09, M’13) received the B.S., M.S., and Ph.D. degrees inelectrical engineering, in 2007, 2009, and 2012, respectively, from the Uni-versity of Illinois, Urbana-Champaign. He is currently an Assistant Professorin the Department of Electrical and Computer Engineering at the Universityof Minnesota (Minneapolis), where he is affiliated with the Power and EnergySystems research group. His research interests include modeling, analysis, andcontrol of power electronics and power systems with a focus on renewableintegration.

Abdullah O. Hamadeh obtained his MEng, MA and PhD degrees inElectrical Engineering from the University of Cambridge in 2005, 2008, and2010 respectively. His doctoral research was in the control and synchronizationof networked dynamical systems. Between 2010 and 2013 he held post-doctoral positions at the University of Waterloo and at Rutgers University.His current research interests are in the applications of control theoretictechniques to systems and synthetic biology and in the control of electricalpower networks.

Philip T. Krein (S’76, M’82, SM’93, F’00) received the B.S. degree inelectrical engineering and the A.B. degree in economics and business fromLafayette College, Easton, Pennsylvania, and the M.S. and Ph.D. degrees inelectrical engineering from the University of Illinois, Urbana.

He was an Engineer with Tektronix in Beaverton, OR, then returned to theUniversity of Illinois at Urbana-Champaign. Currently, he holds the GraingerEndowed Director’s Chair in Electric Machinery and Electromechanics asProfessor and Director of the Grainger Center for Electric Machinery andElectromechanics, Department of Electrical and Computer Engineering. Hepublished an undergraduate textbook Elements of Power Electronics (Oxford,U.K.: Oxford Univ. Press, 1998). In 2001, he helped initiate the IEEEInternational Future Energy Challenge, a major student competition involvingfuel cell power conversion and energy efficiency. He holds twenty six U.S.patents with additional patents pending. His research interests address allaspects of power electronics, machines, drives, and electrical energy, withemphasis on nonlinear control and distributed systems.

Dr. Krein is a Registered Professional Engineer in the States of Illinoisand Oregon. He was a senior Fulbright Scholar at the University of Surrey,Surrey, U.K. and was recognized as a University Scholar, the highest researchaward at the University of Illinois. In 2003, he received the IEEE William E.Newell Award in Power Electronics. He is a past president of the IEEE PowerElectronics Society, and served as a member of the IEEE Board of Directors.In 2005-2007, he was a Distinguished Lecturer for the IEEE Power ElectronicsSociety. In 2008, he received the Distinguished Service Award from the IEEEPower Electronics Society. He is an Associate Editor of the IEEE Transactionson Power Electronics and serves as Academic Advisor for the Department ofElectronic and Information Engineering at Hong Kong Polytechnic University.He is a founder and Director of SolarBridge Technologies, a developer oflong-life integrated solar energy systems.