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Synchronization and Transient Stability in PowerNetworks and Non-Uniform Kuramoto Oscillators

Florian Dorfler, Member, IEEE, and Francesco Bullo, Fellow, IEEE

Abstract

Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss thesynchronization problem for the network-reduced model of a power system with non-trivial transfer conductances.Our key insight is to exploit the relationship between the power network model and a first-order model of coupledoscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbationanalysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols,we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditionsreduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singularperturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronizationand transient stability of a power network to the underlying system parameters and initial conditions.

I. INTRODUCTION

The “smart grid” is a topic that is increasingly gaining interest and importance in politics, science,and technology. The vast North American interconnected power grid is often referred to as the largestand most complex machine engineered by humankind. Obviously, there are various instabilities arising insuch a large-scale power grid, which can be classified by their physical nature, the size of the uncertaintyor disturbance causing the instability, or depending on the devices, processes, and the time necessary todetermine the instability [2]. All of these instabilities can lead and have led to blackouts of power grids [3].The envisioned future power generation will rely increasingly on renewable energy sources such as solarand wind power. Since these renewable power sources are highly stochastic, there will be an increasingnumber of transient disturbances acting on a power grid that is expected to be even more complex anddecentralized than the current one. Thus the detection and rejection of such instabilities will be one ofthe major challenges faced by the future “smart power grid.”

One form of power network stability is the so-called transient stability [4], which is the ability of a powersystem to remain in synchronism when subjected to large transient disturbances. These disturbances mayinclude faults on transmission elements or loss of load, loss of generation, or loss of system componentssuch as transformers or transmission lines. For example, a recent major blackout in Italy in 2003 wascaused by tripping of a tie-line and resulted in a cascade of events leading to the loss of synchronism ofthe Italian power grid with the rest of Europe [5]. The mechanism by which interconnected synchronousmachines maintain synchronism is a balance of their mechanical power inputs and their electrical poweroutputs, which are nonlinearly dependent on the relative rotor angles among machines. In a classic settingthe transient stability problem is posed as a special case of the more general synchronization problem,which is defined over a possibly longer time horizon, for rotor angles possibly drifting away from theirnominal values, and for generators subject to local excitation controllers aiming to restore synchronism. Inorder to analyze the stability of a synchronous operating point of a power grid and to estimate its region ofattraction, various sophisticated algorithms have been developed [6], [7], especially for lossy transmission

This work was supported in part by NSF grants CMS-0626457 and CNS-0834446. This document is a vastly revised and extended versionof [1].

Florian Dorfler and Francesco Bullo are with the Center for Control, Dynamical Systems and Computation, University of California atSanta Barbara, Santa Barbara, CA 93106, dorfler, bullo@engineering.ucsb.edu

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lines with non-trivial transfer conductances [8], [9], [10], [11], [12]. Reviews and survey articles ontransient stability analysis can be found in [13], [14], [15]. Unfortunately, the existing methods can copeonly with physically simplified models and do not provide simple formulas to check if a power systemsynchronizes for a given system state and network parameters. In fact, it is outstanding problem to relatetransient stability of a power network to the actual underlying network structure [16]. From a network-theoretic viewpoint, the transient stability problem is concerned with the robustness of synchronization inface of disturbances in the network’s parameters, state, and topology.

The recent years have witnessed a burgeoning interest of the control community in cooperative control ofautonomous agent systems. Recent surveys and monographs include [17], [18], [19]. One of the basic tasksin a multi-agent system is a consensus of the agents’ states to a common value. This consensus problemhas been subject to fundamental research [20], [21] as well as to applications in robotic coordination,distributed sensing and computation, and various other fields including synchronization. In most articlestreating consensus problems the agents obey single integrator dynamics, but the synchronization ofinterconnected power systems has often been envisioned as possible future application [22]. However,we are only aware of one article [23] that indeed applies consensus methods to a physical power systemmodel.

Another set of literature relevant to our investigation is the synchronization of coupled oscillators.A classic model of coupled oscillators was introduced in the seminal work by Kuramoto [24]. Thesynchronization of coupled Kuramoto oscillators has been widely studied by the physics [25], [26], [27]and the dynamical systems communities [28], [29]. This vast literature with numerous theoretical resultsand rich applications to various scientific areas is elegantly reviewed in [30], [31]. Recent works in thecontrol community [20], [21], [32], [33] investigate the close relationship between Kuramoto oscillatorsand consensus networks.

The three areas of power network synchronization, Kuramoto oscillators, and consensus protocols areapparently closely related, but the gap between the first and the second two topics has not been bridgedyet. Indeed, the similarity between the Kuramoto oscillators model and the power network models used intransient stability analysis is striking. Even though power networks have often been referred to as systemsof coupled oscillators, the similarity to a second-order Kuramoto-type model has been mentioned onlyvery recently in the power networks community in [34], [35], where only qualitative simulation studies arecarried out. In the coupled-oscillators literature, second-order Kuramoto models similar to power networkmodels have been analyzed in simulations and in the continuous limit as partial differential equations; see[31] and references therein. However, we are aware of only one article referring to power networks aspossible application [36]. In short, neither the Kuramoto nor the power systems literature has recognizedand thoroughly analyzed this similarity.

There are three main contributions in the present paper. As a first contribution, we present a coupled-oscillator approach to the problem of synchronization and transient stability in power networks. Via asingular perturbation analysis, we show that the transient stability analysis for the classic swing equationswith overdamped generators reduces, on a long time-scale, to the problem of synchronizing non-uniformKuramoto oscillators with multiple time constants, non-homogeneous coupling, and non-uniform phase-shifts. This reduction to a non-uniform Kuramoto model is arguably the missing link connecting transientstability analysis to networked control, a link that has been hinted at in [16], [22], [34], [35].

Second, we give novel, simple, and purely algebraic conditions that are sufficient for synchronizationand transient stability of a power network. To the best of our knowledge these conditions are the firstones to relate synchronization and performance of a power network directly to the underlying networkparameters and initial state. Our conditions are based on different and possibly less restrictive assumptionsthan those obtained by classic analysis methods [6], [7], [8], [9], [10], [11], [12]. As mathematical modelof a power network we consider a network-reduction model of arbitrary size. In comparison to classicanalysis methods, we do not make any of the common or classic assumptions: we do not require uniformmechanical damping, we do not require the swing equations to be formulated in relative coordinates,we do not require the existence of an infinite bus, and we do not require the transfer conductances to

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be “sufficiently small” or even negligible. On the other hand, our results are based on the assumptionthat each generator is strongly overdamped, possibly due to internal excitation control. This assumptionallows us to perform a singular perturbation analysis and study a reduced-dimension system. In simulationsstudies, our synchronization conditions appear to hold even if generators are not overdamped, and in theapplication to real power networks the approximation via the dimension-reduced system has been usedsuccessfully in academia and industrial practice.

Our synchronization conditions are based on an analytic network-based approach whereas classicanalysis methods [6], [7], [8], [9], [10], [11], [12] rely on numerical procedures to approximate the region ofattraction of a synchronous equilibrium. Compared to classic analysis methods our analysis does not aim atproviding best estimates of the region of attraction or the critical clearing time by numerical computationof stable manifolds of unstable equilibria and level sets of energy functions. Rather, we approach theoutstanding problem [16] of relating synchronization and transient stability to the underlying networkstructure. For this problem, we derive sufficient and purely algebraic conditions. Our sufficient conditionscan be interpreted as follows: “the network connectivity has to dominate the initial lack of phase locking,the network’s non-uniformity, and the network’s losses due to transfer conductances.”

Third and final, we perform a synchronization analysis of non-uniform Kuramoto oscillators, as aninteresting mathematical problem in its own right. Our analysis extends the results obtained for classicKuramoto oscillators and is based on combining and extending tools from transient stability analysis,consensus protocols, and recent results concerning the Kuramoto model. As an outcome, purely alge-braic conditions on the network parameters and the system state establish the phase locking, frequencyentrainment, and phase synchronization of the non-uniform Kuramoto oscillators with and without phaseshifts. We emphasize that our results hold not only for non-uniform network parameters but also in thecase when the underlying topology is not a complete graph. When our results are specialized to classic(uniform) Kuramoto oscillators, they reduce to and even improve upon various well-known conditionsin the literature on Kuramoto oscillators [32], [33], [25], [37]. In the end, these conditions guaranteeingsynchronization of the non-uniform Kuramoto oscillators also suffice for the transient stability of thepower network.

Paper Organization: This article is organized as follows. The remainder of this section introducessome notation, recalls preliminaries on algebraic graph theory, and reviews the consensus protocol andthe Kuramoto model of coupled oscillators. Section II reviews the problem of transient stability analysis.Section III introduces the non-uniform Kuramoto model and presents the main result of this article.Section IV translates the power network model considered in transient stability analysis to the non-uniform Kuramoto model whose synchronization analysis is presented in Section V. Section VI providessimulation studies to illustrate the analytical results. Finally, some conclusions are drawn in Section VII.

Preliminaries and Notation: Given an n-tuple (x1, . . . , xn), diag(xi) ∈ Rn×n is the associated diagonalmatrix, x ∈ Rn is the associated column vector, and xmax and xmin are the maximum and minimumelements, and ‖x‖2 and ‖x‖∞ are the 2- and ∞-norm. The vectors 1 and 0 are the vectors of 1’s and0’s of appropriate dimension. Given two non-zero vectors x ∈ Rn and y ∈ Rn, the angle between them,denoted by ∠(x, y) ∈ [0, π/2], satisfies cos(∠(x, y)) = xTy/(||x|| ||y||). Given a two dimensional arrayAij with i ∈ 1, . . . , n and j ∈ 1, . . . ,m, we let A ∈ Rn×m denote the associated matrix and wedefine Amax = maxi,jAij Amin = mini,jAij. Given a total order relation among the elements Aij , welet diag(Aij) denote the corresponding diagonal matrix. For a differentiable function f : Rn → R, let∇f be the column vector of partial derivatives ∂f/∂xi. To avoid clutter, the upper index n in summationsymbols

∑ni=1 and product symbols

∏ni=1 is avoided whenever it is clear from the context. The set

T1 = (−π, π] is the torus and the product set Tn is the n-dimensional torus. Given two angles θ1 ∈ T1

and θ2 ∈ T1 we define their distance |θ1 − θ2|, with slight abuse of notation, to be the geodesic distanceon T1, as the reader would intuitively assume. Finally, we define the multivariable sine sin : Tn → [0, 1]n

by sin(x) = (sin(x1), . . . , sin(xn)) and the sinc funtion sinc : R→ R by sinc(x) = sin(x)/x.A weighted directed graph is a triple G = (V , E , A), where V = 1, . . . , n is the set of nodes,E ⊂ V × V is the set of directed edges, and A ∈ Rn×n is the adjacency matrix. The entries of A satisfy

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aij > 0 for each directed edge (i, j) ∈ E and are zero otherwise. Any nonnegative matrix A induces aweighted directed graph G. The Laplacian of G is the matrix L(G) = L(aij) := diag(

∑j aij)−A ∈ Rn×n.

The set i, j refers to the pair of nodes connected by either (i, j) or (j, i). In the following, we assumethat A = AT , that is, G is topologically undirected and symmetrically weighted. In this case, the graph Gis fully described by the elements aij with i ≥ j. If a number k ∈ n(n− 1)/2 and a weight wk = aij isassigned to any of these edges (i, j) with i > j, then the incidence matrix H ∈ R(n(n−1)/2)×n is definedcomponent-wise as Hkl = 1 if node l is the sink node of edge k and as Hkl = −1 if node l is thesource node of edge k; all other elements are zero. The Laplacian equals then the symmetric matrixL(G) = HT diag(wk)H . If G is connected, then ker(H) = ker(L(G)) = span(1), and all n− 1 remainingnon-zero eigenvalues of L(G) are strictly positive. The Laplacian’s second-smallest eigenvalue λ2(L(G))is called the algebraic connectivity of G and, for a complete and uniformly weighted graph (aij ≡ 1 forall i and j), it satisfies λ2(L(G)) = n.

Review of the Consensus Protocol and the Kuramoto Model: In a system of n autonomous agents,each characterized by a state variable xi ∈ R, one of the most basic tasks is to achieve a consensus ona common state value. In other words, we aim to achieve that, for all agent pairs i, j the differencevariable xi(t)− xj(t)→ 0 as t→∞. Given a graph with adjacency matrix A describing the interactionbetween agents, a simple linear continuous time algorithm to achieve consensus on the agents’ state isthe consensus protocol

xi = −∑

jaij(xi − xj), i ∈ 1, . . . , n . (1)

In vector notation the consensus protocol (1) takes the form x = −L(aij)x = −L(G)x, which directlyreveals the dependence of the consensus protocol to the underlying graph G.

A well-known and widely used model for the synchronization among coupled oscillators is the Kuramotomodel, which considers n coupled oscillators with state θi ∈ T1 with the dynamics

θi = ωi −K

n

∑j 6=i

sin(θi − θj), i ∈ 1, . . . , n , (2)

where K is the coupling strength and ωi is the natural frequency of oscillator i.

II. MODELS AND PROBLEM SETUP IN SYNCHRONIZATION AND TRANSIENT STABILITY ANALYSIS

A. The Mathematical Model of a Power NetworkIn a power network with n generators we associate with each generator i its internal voltage level Ei > 0,

its self-conductance Gii > 0, and its active power output Pe,i as electrical parameters. Its mechanicalparameters are its mechanical power input Pm,i > 0 from the prime mover, its inertia constant1 Mi > 0,its damping constant Di > 0, and its rotor angle θi measured with respect to a synchronously rotatingreference frame with frequency f0. All these parameters are given in per unit system, except for Mi andDi which are given in seconds, and the synchronous frequency f0 typically given as 50 Hz or 60 Hz.Newton’s second law for the rotor dynamics of generator i leads then to the classic constant-voltagebehind reactance model of interconnected swing equations [13], [38], [39]:

Mi

πf0θi = Pm,i − E2

iGii −Diθi − Pe,i, i ∈ 1, . . . , n .

Under the common assumption that the loads are modeled as passive admittances, all passive nodes of apower network can be eliminated (Kron reduction) resulting in the symmetric reduced admittance matrixY ∈ Cn×n. Its off-diagonal components are Yij = Gij +

√−1Bij for i 6= j, where Gij ≥ 0 and Bij > 0

1In the power system literature the standard variable for the normalized inertia constant is Hi. We refrain from this notation since thevariable H also represents the incidence matrix of a graph.

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are the conductance between generator i and j, respectively the susceptance, in per unit values. With thehighly nonlinear power-angle relationship, the electrical output power is then

Pe,i =∑

j 6=iEiEj (Gij cos(θi − θj) +Bij sin(θi − θj)) .

Given the admittance Yij = Yji between generator i and j, define the magnitude |Yij| =√G2ij +B2

ij > 0

and the phase shift ϕij = arctan(Gij/Bij) ∈ [0, π/2). Recall that energy is lost between two generatorswhenever the corresponding phase shift is strictly positive and that a lossless power network is character-ized by zero phase shifts between all generator pairs. The power network model can then be formulatedcompactly as

Mi

πf0θi = Pm,i − E2

iGii −Diθi −∑

j 6=iEiEj |Yij| sin(θi − θj + ϕij), i ∈ 1, . . . , n . (3)

Typically, a dynamical model for the internal voltage of generator i is given as

Ei = Ei(Ei, ui, θi − θj) ,

where ui is the field excitation and can be used as a control input to the generator [40]. Note that higherorder electrical and flux dynamics in damper windings can be reduced accurately into an augmenteddamping term Di in (3) via integral manifolds [41]. The generator’s internal excitation control via apower system stabilizer essentially increases the damping torque towards the net frequency and can alsobe reduced into the damping term Di [38], [39]. It is commonly agreed that the classic model (3) capturesthe power system dynamics sufficiently well during the first swing [13], [14], [38], [39]. Thus we omithigher order dynamics and control effects and assume they are incorporated into the classic model (3).We remark that all results in this article are also valid if Ei = Ei(t) is a smooth, bounded, and strictlypositive time-varying parameter.

Concluding this section, the power system model (3) is also referred to as a network-reduction modelbecause the power network is reduced to its active nodes. This renders the mathematical model tractablebut structural information about the topology of the original physical power network is almost inherentlylost. Alternatively, network-preserving models can be considered that are systems of differential algebraicequations. These models preserve the original network topology and are subject of the authors’ ongoingresearch.

B. Synchronization and Equilibrium in Power NetworksA frequency equilibrium of (3) is characterized by θ = 0 and by the power flow equations

Qi(θ) = Pm,i − E2iGii − Pe,i ≡ 0, i ∈ 1, . . . , n . (4)

Clearly, system (3) cannot be in frequency equilibrium if there is an unbalance of the mechanical inputpower and the electrically dissipated power in (4). This should not be confused with synchronization:the single generators are said to be in synchronous equilibrium, if the phase differences θi − θj areconstant, respectively the frequency differences θi − θj are zero. We say the power network synchronizes(exponentially) if the phase differences θi(t)−θj(t) become bounded and the frequency differences θi(t)−θj(t) converge to zero (with exponential decay rate) as t → ∞. In the literature on coupled oscillatorsthis is also referred to as phase locking and frequency entrainment, and the case θi = θj for all i, j isreferred to as phase synchronization.

In order to reformulate the synchronization problem as a stability problem, system (3) is usually formu-lated in relative coordinates [42], for instance, in the center of angle coordinates θi− (

∑jMjθj/

∑jMj),

or with machine n as reference in the coordinates θi− θn, i ∈ 1, . . . , n− 1. Unfortunately, the resultingdynamics under these approaches are not self-contained dynamical systems. A common assumptionto circumvent this obstacle is that of uniform damping, that is, Di/Mi constant for all generators i.

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Alternatively, sometimes the existence of an infinite bus as reference is postulated, that is, a referencegenerator, say n, has no dynamics, unit voltage En ≡ 1, and zero angle θn ≡ 0 [7], [6], [14]. Both ofthese assumptions are mathematical simplifications that reduce the synchronization problem to a stabilityanalysis of the origin. We remark that in general these assumptions are not physically justified whenconsidering the synchronization problem of generators with non-identical frequencies.

C. Review of Classic Transient Stability AnalysisClassically, transient stability analysis deals with a special case of the synchronization problem, namely

the stability of a post-fault frequency equilibrium point, that is, a new frequency equilibrium of (3) arisingafter a change in the network parameters or topology. Of special interest is whether, after a fault clearance,the initial condition of the post-fault system will converge to a new stable equilibrium point (θ∗,0). Toanswer this question various sophisticated analytic and numeric methods have been developed [13], [14],[15], which typically employ the gradient-structure of system (3). Since in general a Hamiltonian functionfor model (3) with non-trivial network conductance (ϕij 6= 0) does not exist [9], early transient stabilityapproaches set the phase shifts ϕij to zero [6], [7]. In this case, the power network model (3) takes form

(M/πf0) θ = −Dθ −∇U(θ) , (5)

where U : (−π, π]n → R is the potential energy given by

U(θ) = −∑

i

(Pm,i − E2

iGii

)(θi − θ∗i )−

∑j 6=i

EiEjBij

(cos(θi − θj)− cos(θ∗i − θ∗j )

). (6)

As mentioned above, system (5) has to be transformed and/or simplified in order to render (5) toa self-contained system that has an equilibrium (θ∗,0). In this case, the energy function (θ, θ) 7→(1/2) θT (M/πf0)θ+U(θ) serves semi-globally as a Lyapunov function. In combination with the invarianceprinciple, we have clearly that system (5) will converge to θ = 0 and the largest invariant zero level set of∇U(θ). In order to estimate the region of attraction of a stable equilibrium θ∗, algorithms such as PEBS[7] or BCU [10] consider the associated gradient flow

θ = −∇U(θ) (7)

as a dimension-reduced system. Then it is a fact that (θ∗) is hyperbolic equilibrium of (7) with ks stableeigenvalues if and only if (θ∗,0) is a hyperbolic equilibrium of (5) with ks stable eigenvalues. Moreover,the regions of attractions of both equilibria are comparable in the sense that they are bounded by thestable manifolds of the same equilibria [10, Theorem 5.7]. For further interesting relationships among thesystems (5) and (7), we refer to [7], [10], [14].

Recently there have also been several developments for lossy power networks with non-trivial transferconductances. For fixed system parameters and “sufficiently small” transfer conductances it is possible toanalyze the power system (3) locally as a smooth perturbation of the lossless system with ϕij = 0 [9],[10]. Other approaches addressing lossy power networks compute numerical energy functions [8] or makeuse of an extended invariance principle [11], [12].

On top of the aforementioned simplifications (uniform damping or infinite bus assumption), all thementioned methods in transient stability analysis are by no means straightforward and do not result inconcise and simple conditions but rather in numerical procedures approximating the stability boundaries bylevel sets of energy functions or stable manifolds of unstable equilibria. In the case of lossy power networksthe cited articles derive their results either for special benchmark problems or for general networks with“sufficiently small” transfer conductances. To the best of our knowledge there exists no result that quantifiesthis smallness of Gij for arbitrary networks. Moreover, from a network perspective the existing algorithmssay nothing about the rate of convergence, robustness, or the region of attraction with respect to thenetwork’s state, its parameters, and topology. The following sections will develop a network approachthat addresses these questions quantitatively via purely algebraic tests.

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III. THE NON-UNIFORM KURAMOTO MODEL AND MAIN SYNCHRONIZATION RESULT

A. The Non-Uniform Kuramoto ModelAs we have already mentioned in the introduction, there is a striking similarity between the power

network model (3) and the Kuramoto model (2), which has neither been recognized nor thoroughlyanalyzed by either of the two communities. To emphasize this similarity, the power network model (3)is rewritten in the language of coupled oscillators. If we define the natural frequency ωi := Pm,i −E2iGii (effective power input to generator i) and the coupling weights Pij := EiEj|Yij| (maximum power

transferred between generators i and j), the power network model (3) is equivalently rewritten as

Mi

πf0θi = ωi −Diθi −

∑j 6=i

Pij sin(θi − θj + ϕij), i ∈ 1, . . . , n . (8)

Next, we define the non-uniform Kuramoto model by

Di θi = ωi −∑

j 6=iPij sin(θi − θj + ϕij), i ∈ 1, . . . , n , (9)

where we assume the parameters satisfy the following ranges: Di > 0, ωi ∈ R, Pij > 0, and ϕij ∈ [0, π/2),for all i, j ∈ 1, . . . , n, j 6= j; by convention, the diagonal values Pii and ϕii are set to zero. System (9)may be regarded as a generalization of the classic Kuramoto model (2) with multiple time-constants Di

and non-homogeneous but symmetric coupling terms Pij and phase-shifts ϕij . The non-uniform Kuramotomodel (9) will serve as a link between the power network model (3), the Kuramoto model (2), and theconsensus protocol (1).

Remark III.1 (Second-order mechanical systems and their first-order approximations:) The non-uniformKuramoto model (9) can be seen as an approximation of the second order system (8) for a small “inertiaover damping ratio” Mi/Di or, more specifically, for a ratio 2Mi/Di much smaller than the net frequency2πf0. Spoken differently, system (9) may be seen a long-time approximation of (8) obtained by a singularperturbation analysis.

The analogy between the non-uniform Kuramoto model (9) and the dimension-reduced gradient system(7) is often studied in classic transient stability analysis to approximate the stability properties of thesecond order system (5) [7], [10], [14]. Both models are of first order, have the same right-hand side,and have also the same equilibria with the same stability properties. Strictly speaking, both models differonly in the time constants Di on the left-hand side. The dimension-reduced system (7) is formulated as agradient-system and is used to study the stability of the equilibria of (7), if they exist. If these equilibriado not exist, system (7) has to be reformulated in relative coordinates. The non-uniform Kuramoto model(9), on the other hand, can be directly used to study synchronization and clearly reveals the underlyingnetwork structure.

B. Main Synchronization ResultWe are now in a position to state our main result, which gives conditions on system parameters and

initial conditions that suffice for the synchronization of the power network model (3) and the non-uniformKuramoto model (9).

Theorem III.2 (Main synchronization result) Consider the power network model (3) and the non-uniform Kuramoto model (9) with effective power inputs ωi := Pm,i − E2

iGii, coupling weights Pij =EiEj|Yij|, phase shifts ϕij , and damping terms Di. Assume that the minimal coupling weight is largerthan a critical value, i.e.,

Pmin > Pcritical :=Dmax

n cos(ϕmax)

(maxi,j

( ωiDi

− ωjDj

)+ max

`

∑k

P`kD`

sin(ϕ`k)

). (10)

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Accordingly, define γmax = arccos(cos(ϕmax)Pcritical/Pmin)− ϕmax taking value in (0, π/2− ϕmax). For γ ∈(0, γmax], define the (non-empty) set of bounded phase differences ∆(γ) := θ ∈ Tn : maxi,j |θi − θj| ≤π/2− γ − ϕmax.For the non-uniform Kuramoto model,

1) phase locking: for every γ ∈ (0, γmax) the set ∆(γ) is positively invariant; and2) frequency entrainment: if θ(0) ∈ ∆(γmax), then the frequencies θi(t) of the non-uniform Kuramoto

oscillators (9) synchronize exponentially to some frequency θ∞ ∈ [θmin(0), θmax(0)].For the power network model with initial conditions satisfying θ(0) ∈ ∆(γmax),

1) approximation error: for any initial frequencies θi(0) there exists a constant ε∗ > 0 such that, ifthe perturbation parameter ε := (Mmax)/(πf0Dmin) < ε∗, then the solution (θ(t), θ(t)) of (3) existsfor all t ≥ 0 and it holds uniformly in t that

θ(t)− θ(t) = O(ε), ∀t ≥ 0 ,

θ(t)−D−1Q(θ(t)) = O(ε), ∀t > 0 ,(11)

where θ(t) is the solution to the non-uniform Kuramoto model (9) with initial condition θ(0) = θ(0)and D−1Q(θ) is the power flow (4) scaled by the inverse damping D−1; and

2) asymptotic approximation error: there exists ε and ϕmax sufficiently small, such that the O(ε)approximation errors in equation (11) converge to zero as t→∞.

We discuss the assumption that the perturbation parameter ε needs to be small separately and in detailin the next subsection and state the following remarks to Theorem III.2:

Remark III.3 (Physical interpretation of Theorem III.2:) The condition (10) on the network parame-ters has a direct physical interpretation when it is rewritten as

nPmin

Dmax

cos(ϕmax) > maxi,j

( ωiDi

− ωjDj

)+ max

`

∑k

P`kD`

sin(ϕ`k) . (12)

The right-hand side of (12) states the worst-case non-uniformity in natural frequencies (the differencein effective power inputs at each generator) and the worst-case lossy coupling of a node to the network(P`k sin(ϕ`k) = E`EkG`k reflects the transfer conductance), both of which are scaled with the rates Di.These negative effects have to be dominated by nPmin cos(ϕmax)/Dmax on the left-hand side of (12), whichis a lower bound on minm

∑kPmk

Dmcos(ϕmk), the worst-case lossless coupling of a node to the network.

The gap between the left- and the right-hand side in (12) determines the worst admissible initial lack ofphase locking in ∆(γmax). In summary, the conditions of Theorem III.2 read as “the network connectivityhas to dominate the initial lack of phase locking, the network’s non-uniformity, and the network’s reallosses.”

The weakest power flow Pmin in condition (10) is not only the minimal coupling weight in the networkbut has also a graphical interpretation as connectivity measure and is related to the physical properties ofthe original (non-reduced) power network: if we fix the voltages Ei and the phase shifts ϕij at constantvalues for all i, j ∈ 1, . . . , n, then, up to some multiplicative constant, P is the all-to-all admittancematrix of the power network and 1/Pmin is the maximum pairwise effective resistance. The effectiveresistance (or resistance distance) is a well-studied graph theoretic concept; a classic reference being [43].It is known, for example, that the algebraic connectivity is upper and lower bounded by Pmin [44, Theorem4.2].

Remark III.4 (Refinement of Theorem III.2 in the lossless case:) Theorem III.2 can be refined in thecase of a lossless network, that is, ϕij = 0 for all i, j. In this case, the worst-case conditions ofTheorem III.2 on the network’s parameters and its initial state can be relaxed, explicit values for thesynchronization frequency and the exponential synchronization rate can be derived, and conditions forphase synchronization can be given (see Theorem V.9).

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Remark III.5 (Reduction of Theorem III.2 to classic Kuramoto oscillators:) When specialized to clas-sic Kuramoto oscillators (2), the presented condition improves the results obtained by [32], [33], [26],[25], [37]. We refer the reader to the detailed remarks in Section V.

Remark III.6 (Sufficiency of Theorem III.2:) The presented condition is only sufficient and in latersimulation studies (in Section VI) we observe that it is a conservative estimate for the region of attraction.However, its main implication “the network connectivity has to dominate the initial lack of phase locking,the network’s non-uniformity, and its real losses” seems to hold in general. Moreover, the condition showsat which generator the damping torque has to be increased or decreased (via local power system stabilizers)in order to meet the sufficient conditions for synchronization.

C. Discussion of the Perturbation AssumptionThis subsection discusses the assumption that each generator is strongly overdamped which is captured

by the smallness of the perturbation parameter ε := (Mmax)/(πf0Dmin). This choice of the perturbationparameter ε and the subsequent singular perturbation analysis (carried out in Section IV) is similar to whatis done in the analysis of Josephson arrays [27], coupled overdamped mechanical pendula [45], and alsoin classic transient stability analysis (in particular Theorem 5.2 in [7]). In the linear case, this singularperturbation analysis resembles the well-known example of the overdamped harmonic oscillator, whichfeatures one slow and one fast eigenvalue. In the extremely overdamped case the harmonic oscillatorthus exhibits two distinct time-scales and the fast eigenvalue corresponding to the frequency dampingcan be neglected in the long-term phase dynamics. In the non-linear case these two distinct time-scales are captured by a singular perturbation analysis with the perturbation parameter ε. In short, thisdimension-reduction of a coupled pendula system corresponds to the physical assumption that dampingand synchronization happen on separate time scales.

In the application to realistic generator models one has to be careful under which operating conditionsε is indeed a small physical quantity. Typically, Mi ∈ [2s, 8s] depending on the type of generator andthe mechanical damping (including damper winding torques) is poor: Di ∈ [1, 2]/(2πf0). However, forthe synchronization problem also the generator’s internal excitation control have to be considered andincrease the damping torque to Di ∈ [10, 35]/(2πf0) depending on the system load [38], [39], [41]. Inthis case, ε ∈ O(0.1) is indeed a small quantity and a singular perturbation approximation is accurate.

We note that in the later simulation studies in Section VI synchronization of both the power networkand the non-uniform Kuramoto model can also be observed for larger values of ε.

The assumption that the quantity ε is small seems to be crucial for the approximation of the powernetwork model by the non-uniform Kuramoto model. However, similar results can also be obtainedindependently of the magnitude of ε. In Remark III.1 we already discussed the similarity between thenon-uniform Kuramoto model (9) and the dimension-reduced system (7) considered in classic transientstability analysis. The transient stability literature has developed different algorithms such as as PEBS [7]and BCU [10] that derive various static and dynamic analogies between the full power network model(3) and the reduced first-order model. These algorithms have been successfully applied in academia andin industrial applications [46]. Among other things, it is found that both models have the same equilibriawith the same local stability properties, and the regions of attractions are also comparable [10, Theorem5.7], as we mentioned before in Section II. These results hold independently of the magnitude of ε andsupport the approximation of the full power network model by the non-uniform Kuramoto model.

IV. SINGULAR PERTURBATION ANALYSIS OF SYNCHRONIZATION

A. Time-Scale Separation of the Power Network ModelIn this section, we put the approximation of the power network model (3) by the non-uniform Kuramoto

model (9) on solid mathematical ground via a singular perturbation analysis. As mentioned above, system(9) may be seen a long-time approximation of (3), or spoken differently, it is the reduced system obtained

10

by a singular perturbation analysis. A physically reasonable singular perturbation parameter is the worst-case choice of Mi/πf0Di, that is,

ε :=Mmax

πf0Dmin.

The dimension of ε is in seconds, which makes sense since time still has to be normalized with respect toε. In order to proceed consider the term Fi := (Di/Dmin)/(Mi/Mmax) , which will determine the speed ofconvergence of the full system (3) to the reduced system (9). With ε, Fi, Pij , and ωi the power networkmodel (3) can be rewritten as the singular perturbation problem

εθi =− Fi θi +FiDi

(ωi −

∑j 6=i

Pij sin(θi − θj + ϕij)), i ∈ 1, . . . , n . (13)

Note that for ε = 0, system (13) reduces the non-uniform Kuramoto model (9) or, if we freeze time, itreduces to a set of algebraic equations in (θ, θ) that can be solved for θi as θi = Qi(θ)/Di, where Qi(θ)is the power flow (4). For ε sufficiently small, the synchronization dynamics of (13) can be approximatedby the non-uniform Kuramoto model (9) and the power flow (4), where the error is of order ε.

Theorem IV.1 (Singular Perturbation Approximation) Consider the power network model (3) writtenas the singular perturbation problem (13) with initial conditions (θ(0), θ(0)) and solution (θ(t, ε), θ(t, ε)).Consider furthermore the non-uniform Kuramoto model (9) as the reduced model with initial conditionθ(0) and solution θ(t), the quasi-steady state h(θ) defined component-wise as hi(θ) = Qi(θ)/Di fori ∈ 1, . . . , n, and the boundary layer error

yi(t/ε) := (θi(0)− hi(θ(0)))e−Fit/ε, i ∈ 1, . . . , n .

Let T > 0 be arbitrary but finite and assume that the initial frequencies θi(0) are bounded.Then, there exists ε∗ > 0 such that for all ε < ε∗, the singular perturbation problem (13) has a unique

solution on [0, T ], and for all t ∈ [0, T ] it holds uniformly in t that

θ(t, ε)− θ(t) = O(ε), and θ(t, ε)− h(θ(t))− y(t/ε) = O(ε) .

Moreover, given any Tb ∈ (0, T ), there exists ε∗ ≤ ε∗ such that for all t ∈ [Tb, T ] and whenever ε < ε∗

θ(t, ε)− h(θ(t)) = O(ε) .

Proof: In the following we let i ∈ 1, . . . , n and omit vector notation as usual in singular perturbationanalysis. If we reformulate system (13) in state space with the state (x, z) = (θ, θ) ∈ Tn × Rn, then weobtain the following system in singular perturbation standard form

xi =fi(z) := zi , (14)

εzi =gi(x, z) := −Fi zi +FiDi

(ωi −

∑j 6=i

Pij sin(xi − xj + ϕij)). (15)

The quasi-steady-state of (14)-(15) is obtained by solving gi(x, z) = 0 for z resulting in the unique rootzi = hi(x). In singular perturbation analysis, we are interested in approximating (14)-(15) by the reducedsystem xi = fi(h(x)) = hi(x), which is equivalent to (9) in x-coordinates. The reduced system is smooth,evolves on the compact set Tn, and thus its solution exists for all t ≥ 0. Let us introduce the error coordinateyi = zi − hi(x), which shifts the error made by the quasi-stationarity assumption zi(t) = hi(x(t)) to theorigin. After stretching time to the dimensionless variable τ = t/ε, the quasi-steady-state error is givenby

d

d τyi = gi(x, y + h(x))− ε∂hi

∂xifi(y + h(x)) = −Fi yi − ε

∂hi∂xi

fi(y + h(x)) (16)

with initial condition yi(0) = zi(0)−hi(x(0)). By setting ε = 0, (16) reduces to the boundary layer modeld

d τyi = −F yi, yi(0) = zi(0)− hi(x(0)) . (17)

11

Note that (17) is globally exponentially stable and its solution depicts the boundary layer error which isequivalent yi(t/ε) in original coordinates. The boundedness of (θ(0), θ(0)) = (x(0), z(0)) guarantees thatyi(0) is in a compact subset of the region of attraction of (17).

In summary, the singularly perturbed system (14)-(15) is smooth, the evolution of the reduced systemxi = fi(h(x)) = hi(x) is bounded, and the boundary layer model (17) is exponentially stable. Thus allassumptions of Tikhonov’s theorem ([47], Theorem 11.1) are satisfied and the theorem follows

The singular perturbation result in Theorem IV.1 may be interpreted geometrically as follows. Thesynchronization of system (13) happens on a slow manifold whose first order approximation is thescaled power flow θ = h(θ) = D−1Q(θ). The dynamics on this slow manifold are given by the non-uniform Kuramoto model (9) and reflect the long-term synchronization behavior of system (13). Thedamping of system (13) happens on a fast time-scale, the boundary layer, and the (initial) error of the fullsystem frequency dynamics versus the slow (quasi-stationary) synchronization manifold is given by theexponentially stable boundary layer model with convergence rate determined by (Di/Dmin)/(Mi/Mmax).

B. Singular Perturbation Analysis on the Infinite Time IntervalAs usual in perturbation and averaging analysis, Theorem IV.1 holds on a finite time interval. In order

to render the O(ε) approximation valid on an infinite time interval, additionally exponential stability ofthe reduced system’s origin is required. Among other things, the following section will show that the non-uniform Kuramoto model synchronizes exponentially under certain conditions on the network’s parametersand its initial state. In this case, we can state the following corollary of Theorem IV.1.

Corollary IV.2 Under the assumption that the non-uniform Kuramoto model (9) synchronizes exponen-tially for some initial condition θ(0), the singular perturbation approximation in Theorem IV.1 is valid onthe infinite time interval, that is, for any T > 0. Moreover, there exist ε and ϕmax sufficiently small suchthat the approximation error converges to zero as t→∞.

Proof: The exponential synchronization of the non-uniform Kuramoto oscillators to some frequencyθi(t)→ θ∞ as t→∞ implies that all phase differences θi(t)− θj(t) converge exponentially to constantvalues. After switching to a rotating frame with angular speed θ∞, it follows that phases θ(t) convergeexponentially to constant values θ∞ := limt→∞(θ(t)− θ∞t1).

Consider now again the power network model in the singular perturbation standard form (14)-(15), andapply the coordinate transformation (x, z) 7→ (x − θ∞ + θ∞t1, z) resulting in a time-invariant system.From the arguments given above, it follows that the origins (in new coordinates) of both the reducedand the boundary layer system are exponentially stable, where Lyapunov functions are readily existent byconverse arguments. Therefore, the singular perturbation approximation in Theorem IV.1 can be extendedto the infinite time interval2 (for any T > 0) [47, Theorem 11.2].

Now we invoke classic arguments from transient stability analysis [10] to show that the O(ε) errorconverges to zero. Both the second order system (14)-(15) as well as the reduced system ˙x = f(h(x))(the non-uniform Kuramoto model) can be formulated with the vector of power flows Q(x) as

D ˙x = Q(x), (18)(M/πf0) x = −Dx+Q(x) . (19)

Let the solution of (18) be termed x(t), and let x(t) and z(t) = x(t) be the solution of (19). By ourprevious argumentation, x(t) converges to the origin, and (x(t), z(t)) converge to an O(ε) neighborhood ofthe origin. Due to Theorem 5.7 in [10], the origin of (19) is also locally exponentially stable for sufficientlysmall phase shifts ϕij . Thus for sufficiently small ε and ϕmax, the solution of (19) also converges to theorigin, and the errors x(t)− x(t) and z(t)− h(x(t)) converge exponentially to zero.

2Note that Theorem III.2 states only a special case of the infinite time interval version of Theorem IV.1, namely the case ε < ε∗ andt ≥ Tb > 0.

12

V. SYNCHRONIZATION OF NON-UNIFORM KURAMOTO OSCILLATORS

In this section we combine tools from transient stability analysis, consensus protocols, and Kuramotooscillators in order to analyze the non-uniform Kuramoto model (9). We remark that system (9) canbe viewed as a multiple-rate and symmetrically but non-homogeneously coupled Kuramoto system withsymmetric phase shifts or as a nonlinear undirected consensus algorithm evolving on Tn. Since the non-uniform Kuramoto model (9) is derived from the power network model (3), the underlying graph inducedby the coupling weights Pij is complete and symmetric, i.e., except for the diagonal entries, the matrix Pis fully populated (all-to-all coupling) and symmetric: Pij = Pji > 0 for all i, j ∈ 1, . . . , n2.

One difficulty in the analysis of system (9) is revealed when dividing by Di both hand sides: the weightscoupling the oscillators i and j are Pij/Di on the one hand and Pij/Dj on the other, that is, the couplingweights are non-symmetric. Additionally, the coupling functions themselves are also non-symmetric sincethe terms sin(θi − θj + ϕij) and sin(θj − θi + ϕij) do not add up to zero. For the sake of generality, thissection considers the non-uniform Kuramoto model (9) under the assumption that the graph induced byP is neither complete nor symmetric, that is, some coupling terms Pij are zero and P 6= P T .

A. Exponential Synchronization of Phase-locked OscillatorsThe frequencies of the classic Kuramoto oscillators (2) synchronize only for locked phase differences,

that is, the phase differences θi(t) − θj(t) must be (ultimately) bounded for all pairs i, j in order toachieve frequency entrainment [32], [33], [25], [37]. Under the assumption of locked phases, we can statean analogous result guaranteeing synchronization of the non-uniform Kuramoto oscillators (9) wheneverthe graph induced by P has a globally reachable node.

Theorem V.1 (Frequency entrainment) Consider the non-uniform Kuramoto model (9) where the graphinduced by P has a globally reachable node. Assume that there exists γ ∈ (0, π/2− ϕmax) such that thenon-empty set of bounded phase differences

∆(γ) = θ ∈ Tn : maxi,j |θi − θj| ≤ π/2− γ − ϕmax

is positively invariant. Then for every θ(0) ∈ ∆(γ) the frequencies θi(t) of the non-uniform Kuramotooscillators (9) synchronize exponentially to some frequency θ∞ ∈ [θmin(0), θmax(0)].

Proof: By differentiating the non-uniform Kuramoto model (9) we obtain the dynamical systemdescribing the evolution of the frequencies

d

d tDiθi =

∑j 6=i

Pij cos(θi − θj + ϕij) (θj − θi) . (20)

Given the matrix P , consider a directed weighted graph G induced by the matrix with elements aij =(Pij/Di) cos(θi − θj + ϕij). Then the frequency dynamics (20) can be written compactly in vector formas the consensus protocol

d

d tθ = −L(aij) θ , (21)

where the matrix L(aij) is a phase-weighted Laplacian corresponding to G. By assumption we have forevery θ(0) ∈ ∆(γ) that θ(t) ∈ ∆(γ) for all t ≥ 0. Consequently the weights aij(t) = (Pij/Di) cos(θi(t)−θj(t) + ϕij) are non-degenerate, i.e., zero for Pij = 0 and otherwise always strictly positive for all t ≥ 0.Note also that system (21) evolves on the tangent space of Tn, that is, the Euclidean space Rn. Therefore,the dynamics (21) can be analyzed as a linear system for the velocities θi, where L(aij) is a state-dependentmatrix, that we analyze as if it was a just time-varying Laplacian matrix.

Note that the weights aij(t) are bounded continuous functions of time and induce integrated over anynon-zero time interval a graph with non-degenerate weights and a globally reachable node. Furthermore,at each time instance the matrix −L(aij(t)) is Metzler with zero row sums. Thus (21) is a linear consensus

13

algorithm for the velocities θi, and it follows from the contraction property [48, Theorem 1] that θi(t) ∈[θmin(0), θmax(0)] for all t ≥ 0, and the dynamics (9) converge exponentially to the set, where θi = θj = θ∞for all i, j ∈ 1, . . . , n2.

For zero phase shifts ϕij = 0 and symmetric coupling terms P = P T , the oscillators (9) are symmet-rically coupled up to the damping terms Di appearing in the weights Pij/Di. In this case, it is possibleto derive an explicit synchronization frequency and give an estimate for the exponential synchronizationrate depending on the network parameters and the initial state. Towards this we borrow the center ofangle formulation from transient stability analysis and introduce the constant weighted mean frequencyΩc defined as

Ωc :=

∑iDiθi∑iDi

=

∑i ωi∑iDi

.

Analogously to the classic Kuramoto oscillators (2) which synchronize to the average frequency (1/n)∑

i ωi,the non-uniform Kuramoto oscillators (9) synchronize to the weighted mean frequency Ωc.

Theorem V.2 (Frequency entrainment for zero phase shifts) Consider the non-uniform Kuramoto model(9) where the graph induced by P = P T is symmetric and connected. Assume there exists γ ∈ (0, π/2)such that the non-empty set of bounded phase differences

∆(γ) = θ ∈ Tn : maxi,j |θi − θj| ≤ π/2− γ

is positively invariant. Then for every θ(0) ∈ ∆(γ) the frequencies θi(t) of the non-uniform Kuramotooscillators (9) synchronize exponentially to the weighted mean frequency Ωc =

∑i ωi/

∑iDi, and the

exponential synchronization rate is no worse than

λfe = −λ2(L(Pij)) sin(γ) cos(∠(D1,1))2/Dmax . (22)

In the definition of the convergence rate λfe in (22), the factor λ2(L(Pij)) is the algebraic connectivityof the graph induced by P = P T , the factor 1/Dmax is the slowest time constant of the non-uniformKuramoto system (9), the proportionality λfe ∼ sin(γ) reflects the phase locking, and the proportionalityλfe ∼ cos(∠(D1,1))2 reflects the fact that the error coordinate θ−Ωc1 is for non-uniform damping termsDi not orthogonal to the agreement vector Ωc1.

Remark V.3 (Reduction of Theorem V.2 to classic Kuramoto oscillators:) Theorem V.2 and its proofare inspired by average consensus protocols [17], [18] and Theorem 3.1 in [32]. When specialized to theclassic Kuramoto oscillators (2) and using the lower bound λ2(L(Pij)) ≥ nPmin for the complete graph,Theorem V.2 reduces to Theorem 3.1 in [32].

Proof of Theorem V.2: Consider the frequency dynamics (20) identified as consensus protocol in theproof of Theorem V.1. For zero shifts the consensus protocol (20) reads as

d

d tDiθi =

∑j 6=i

Pij cos(θi − θj) (θj − θi) . (23)

For each coupling term Pij 6= 0 define the strictly positive weight wij := Pij cos(θi − θj) > 0 dependingon the phases θ(t) ∈ ∆(γ). Let H be the incidence matrix of the symmetric graph induced by P = P T ,and consider the symmetric phase-weighted Laplacian matrix L(wij) = HT diag(wij)H . This allows usto rewrite (23) compactly as a symmetric consensus protocol with multiple rates D:

d

d tDθ = −L(wij) θ . (24)

From Theorem V.1 it follows that the oscillators synchronize exponentially to some frequency θ∞. SinceL(wij) is symmetric, it holds that 1T d

d tDθ = 0, or equivalently,

∑iDiθi(t) is a constant conserved

14

quantity. If we apply this argument again at θ∞ := limt→∞ θi(t), then we have∑

iDiθi(t) =∑

iDiθ∞,or equivalently, the frequencies synchronize exponentially to θ∞ = Ωc.

In order to derive an explicit synchronization rate to Ωc, we introduce the weighted disagreement vectorδ = θ − Ωc1, as an error coordinate satisfying

1TDδ = 1TDθ − 1TDΩc1 =∑

iDiθi −

∑iDiθi = 0 ,

that is, δ lives in the weighted disagreement eigenspace of dimension n − 1 and with normal vectorD 1. Since Ωc is constant and ker(L(wij)) = span(1) (due to connectivity), the dynamics (21) written inδ-coordinates simplify to the weighted disagreement dynamics

d

d tDδ = −L(wij) δ . (25)

Consider the weighted disagreement function δ 7→ δTDδ and its derivative along the disagreementdynamics (25) which is

d

d tδTDδ = −2 δTL(wij)δ .

Note that δ is a vector in the weighted disagreement eigenspace, that is, δTD1 = 0. Therefore, in generalδ 6∈ span(1) and δ can be uniquely decomposed into orthogonal components as δ = (1T δ/n)1 + δ⊥,where δ⊥ is the orthogonal projection of δ on the subspace orthogonal to 1. By the Courant-FischerTheorem [49], the time derivative of the weighted disagreement function can be upper-bounded with thesecond-smallest eigenvalue of the Laplacian L(wij), i.e., the algebraic connectivity λ2(L(wij)), as

d

d tδTDδ ≤ −2λ2(L(wij)) ‖δ⊥‖2 .

Since the Laplacian can be written as L(wij) = HT diag(Pij cos(θi − θj))H , the algebraic connectivityλ2(L(wij)) can be lower-bounded as

λ2(L(wij)) ≥ λ2(L(Pij)) mini,jcos(θi − θj) : θ ∈ ∆(γ) ≥ λ2(L(Pij)) sin(γ) .

In the sequel, we will show that ‖δ⊥‖ may be bounded by ‖δ‖. In order to do so, we let 1⊥ = (1/ ‖δ⊥‖) δ⊥be the unit vector that δ is projected on (in the subspace orthogonal to 1). The norm of δ⊥ can be obtainedas ‖δ⊥‖ =

∥∥δT1⊥∥∥ = ‖δ‖ cos(∠(δ,1⊥)). The vectors δ and 1⊥ each live on (n − 1)-dimensional linearhyperplanes with normal vectors D1 and 1, respectively, see Figure 1 for an illustration. The angle∠(δ,1⊥) is upper-bounded by maxδ ∠(δ,1⊥), which is said to be the dihedral angle and its sine is thegap between the two subspaces [49]. Since both hyperplanes are of co-dimension 1, we obtain the dihedralangle as the angle between the normal vectors D1 and 1, and it follows that ∠(δ,1⊥) ≤ ∠(D1,1) (withequality for n = 2). In summary, we have ‖δ‖ ≥ ‖δ⊥‖ ≥ ‖δ‖ cos(∠(D1,1)).

1D1

!

!!

!(D1,1)

Fig. 1. Qualitative illustration of the disagreement eigenspace and the orthogonal complement of 1

15

Finally, given Dmin ‖δ‖2 ≤ δTDδ ≤ Dmax ‖δ‖2 and λfe as stated in equation (22), we obtain for thederivative of the disagreement function

d

d tδTDδ ≤ −2λfeδ

TDδ .

An application of the Bellman-Gronwall Lemma yields δ(t)TDδ(t) ≤ δ(0)TDδ(0) e−2λfe(t) for all t ≥ 0.Alternatively, after reusing the bounds on δTDδ, we obtain that the disagreement vector δ(t) satisfies‖δ(t)‖ ≤

√Dmax/Dmin ‖δ(0)‖ e−λfe(t) for all t ≥ 0.

Remark V.4 (Synchronization frequency in the case of non-zero phase shifts:) Theorem V.2 allowsus also to state a qualitative result on the synchronization frequency θ∞ in the case of symmetric couplingP = P T and non-zero phase-shifts ϕij 6= 0. A small signal analysis of the non-uniform Kuramoto model(9) with ϕij 6= 0 reveals that the natural frequency of each oscillator decreases as ωi −

∑j 6=i Pijϕij .

Consequently, the synchronization frequency θ∞ in the lossy case will be smaller than Ωc.

B. Phase LockingThe key assumption in Theorem V.1 and Theorem V.2 is that phase differences are bounded in the

set ∆(γ). This subsection provides two different approaches to deriving conditions for this phase lockingassumption - the contraction property and ultimate boundedness arguments in case of zero phase shifts.

The dynamical system describing the evolution of the phase differences for the non-uniform Kuramotomodel (9) reads component-wise as

θi − θj =ωiDi

− ωjDj

−∑

k

(PikDi

sin(θi − θk + ϕik)−PjkDj

sin(θj − θk + ϕjk)

), (26)

where the summation index k can be evaluated in 1, . . . , n since Pii = Pjj = 0. In order to show thephase locking, the Kuramoto literature provides various methods such as quadratic Lyapunov functions[32], contraction mapping [33], geometric [25], linearization [29], or Hamiltonian arguments [29], [28],[26]. Due to the non-uniform and non-symmetric coupling via the weights Pij/Di and the phase-shiftsϕij none of the mentioned methods appears to be easily applicable to the non-uniform Kuramoto model.A different approach from the literature on consensus protocols [20], [21], [50], [37], [51], [50] is basedon convexity and contraction and aims to show that the arc in which all phases are contained is of non-increasing length. A modification of this approach turns out to be applicable to non-uniform Kuramotooscillators with a complete graph and results in the following theorem.

Theorem V.5 (Synchronization) Consider the non-uniform Kuramoto-model (9) and the phase differ-ences system (26), where the graph induced by P = P T is symmetric and complete. Assume that theminimal coupling is larger than a critical value, i.e.,

Pmin > Pcritical :=Dmax

n cos(ϕmax)

(maxi,j

( ωiDi

− ωjDj

)+ max

`

∑k

P`kD`

sin(ϕ`k)

). (27)

Accordingly, define γmax = arccos(cos(ϕmax)Pcritical/Pmin)− ϕmax taking value in (0, π/2− ϕmax). For γ ∈(0, γmax], define the (non-empty) set of bounded phase differences ∆(γ) = θ ∈ Tn : maxi,j |θi − θj| ≤π/2− γ − ϕmax.For the non-uniform Kuramoto model,

1) phase locking: for every γ ∈ (0, γmax) the set ∆(γ) is positively invariant; and2) frequency entrainment: for every θ(0) ∈ ∆(γmax) the frequencies θi(t) of the non-uniform Ku-

ramoto oscillators (9) synchronize exponentially to some frequency θ∞ ∈ [θmin(0), θmax(0)].

Condition (27) is interpreted in Remark III.3. In essence, Theorem V.5 is based on the contractionproperty: the positive invariance of ∆(γ) is equivalent to showing that all angles θi ∈ T are contained in

16

a rotating arc of non-increasing maximal length π/2 − γ − ϕmax. This contraction analysis is similar tothat of the consensus and rendezvous algorithms in [20], [21], [37], [50], which derive their results onRn.

Remark V.6 (Reduction of Theorem V.5 to classic Kuramoto oscillators:) For the classic Kuramotooscillators (2) the sufficient condition (27) of Theorem V.5 specializes to

K > Kcritical := ωmax − ωmin . (28)

In other words, if K > Kcritical, then there exists a positive-measure set of initial phase differences∆(arccos(Kcritical/K)), such that the oscillators synchronize. To the best of our knowledge, the con-dition (28) on the coupling gain K is the tightest bound sufficient for synchronization that has beenpresented in the Kuramoto literature so far. In fact, the bound (28) is close to the necessary conditionsfor synchronization derived in [32], [33], [26]. For example, the necessary condition in [32], [33] readsK > Kcritical n/(2(n− 1)). Thus, in the case of two oscillators, condition (28) is necessary and sufficientfor the onset of synchronization.

Other sufficient bounds given in the Kuramoto literature scale asymptotically with n, e.g., [33, Theorem2] or [32, proof of Theorem 4.1]. To compare our condition (28) with the bounds in [32], [25], [37], wenote from the proof of Theorem V.5 that our condition can be equivalently stated as follows. The set ofbounded phase differences ∆(γ), for γ ∈ (0, π/2), is positively invariant if

K ≥ K(γ) :=Kcritical

cos(γ)=ωmax − ωmin

cos(γ). (29)

Our bound (29) improves the bound K > K(γ)n/2 derived in [32, proof of Theorem 4.1] via a quadraticLyapunov function and the bound K > K(γ)n/(n− 2) derived in [37] via contraction arguments similarto ours. Additionally, our bound (29) strictly improves the bound derived geometrically in [25, proof ofProposition 1] that, after some manipulations, can be written in our notation as K ≥ K(γ) cos((π/2 −γ)/2)/ cos(π/2 − γ). In summary, the bound (27) in Theorem V.5 improves the known [32], [33], [26],[25], [37] sufficient conditions for synchronization of classic Kuramoto oscillators, and is a necessary andsufficient condition in the case of two oscillators.

Proof of Theorem V.5: We start by proving the positive invariance of ∆(γ). Recall the geodesic distanceamong two angles on the torus T1 and define the non-smooth function V : Tn → [0, π] by

V (ψ) = max|ψi − ψj| | i, j ∈ 1, . . . , n.

A simple geometric argument shows the following fact: if V (ψ) < 2π/3, then there exists a uniquearc (i.e., a connected subset of T1) of length V (ψ) containing ψ1, . . . , ψn. This arc has two boundarypoints: a counterclockwise maximum and a counterclockwise minimum. If we let Imax(ψ) (respectivelyImin(ψ)) denote the set indices of the angles ψ1, . . . , ψn that are equal to the counterclockwise maximum(respectively the counterclockwise minimum), as shown in Figure 2, then we may write

V (ψ) = |ψm′ − ψ`′ |, for all m′ ∈ Imax(ψ) and `′ ∈ Imin(ψ).

By assumption, the angles θi(t) belong to the set ∆(γ) at time t = 0. We aim to show that they remainso for all subsequent times t > 0. Note that θ(t) ∈ ∆(γ) if and only if V (θ(t)) ≤ π/2−γ−ϕmax < 2π/3.Therefore, ∆(γ) is positively invariant if and only if V (θ(t)) does not increase at any time t such thatV (θ(t)) = π/2 − γ − ϕmax. The upper Dini derivative of V (θ(t)) along the dynamical system (26) isgiven as [21, Lemma 2.2]

D+V (θ(t)) = limh↓0

supV (θ(t+ h))− V (θ(t))

h= θm(t)− θ`(t) ,

17

Imin(!)Imax(!)

V (!)

Fig. 2. Illustration of the ingredients of the contraction analysis on T

where m ∈ Imax(θ(t)) and ` ∈ Imin(θ(t)) are indices with the properties that

θm(t) = maxθm′(t) | m′ ∈ Imax(θ(t)), and θ`(t) = minθ`′(t) | `′ ∈ Imin(θ(t)).Written out in components D+V (θ(t)) takes the form

D+V (θ(t)) =ωmDm

− ω`D`

−∑

k

(PmkDm

sin(θm(t)− θk(t) + ϕmk)−P`kD`

sin(θ`(t)− θk(t) + ϕ`k)

).

Adding and subtracting the coupling with zero phase shifts yields

D+V (θ(t)) =ωmDm

− ω`D`

−∑

k

(PmkDm

sin(θm(t)− θk(t)) +P`kD`

sin(θk(t)− θ`(t)))

−∑

k

PmkDm

(sin(θm(t)− θk(t) + ϕmk)− sin(θm(t)− θk(t))

)+∑

k

P`kD`

(sin(θ`(t)− θk(t)) + ϕ`k) + sin(θk(t)− θ`(t))

).

Since both sinusoidal terms in the first sum are strictly positive, they can be lower-bounded asPmkDm

sin(θm(t)− θk(t)) +P`kD`

sin(θk(t)− θ`(t)) ≥Pmin

Dmax

(sin(θm(t)− θk(t)) + sin(θk(t)− θ`(t))

).

In the following we apply classic trigonometric arguments from the Kuramoto literature [32], [25], [37].The identity sin(x) + sin(y) = 2 sin(x+y

2) cos(x−y

2) leads to the further simplifications

sin(θm(t)− θk(t)) + sin(θk(t)− θ`(t)) = 2 sin

(θm(t)− θ`(t)

2

)cos

(θm(t) + θ`(t)

2− θk(t)

),

sin(θm(t)− θk(t) + ϕmk) + sin(θk(t)− θm(t)) = 2 sin(ϕmk

2

)cos(θm(t)− θk(t) +

ϕmk2

),

sin(θ`(t)− θk(t) + ϕ`k) + sin(θk(t)− θ`(t)) = 2 sin(ϕ`k

2

)cos(θ`(t)− θk(t) +

ϕ`k2

).

Now, the equality V (θ(t)) = π/2 − γ − ϕmax implies that, measuring distances counterclockwise andmodulo additional terms equal to multiples of 2π,

θm(t)− θ`(t) = π/2− γ − ϕmax,

0 ≤ θm(t)− θk(t) ≤ π/2− γ − ϕmax,

0 ≤ θk(t)− θ`(t) ≤ π/2− γ − ϕmax.

Therefore, D+V (θ(t)) may be upper bounded as

D+V (θ(t)) ≤ maxi,j

( ωiDi

− ωjDj

)− Pmin

Dmax

∑k

2 sin(π/2− γ − ϕmax

2

)cos(π/2− γ − ϕmax

2

)−∑

k

PmkDm

2 sin(ϕmk

2

)cos(π/2− γ − ϕmax +

ϕmk2

)+∑

k

P`kD`

2 sin(ϕ`k

2

)cos(ϕ`k

2

).

18

Note that the second sum is strictly negative and can be neglected. Moreover, in the third sum the maximumover all nodes l can be taken. Reversing the trigonometric identity from above as 2 sin(x) cos(y) =sin(x− y) + sin(x+ y) yields then the simple expression

D+V (θ(t)) ≤ maxi,j

( ωiDi

− ωjDj

)− Pmin

Dmax

∑k

cos(γ + ϕmax) + max`

∑k

P`kD`

sin(ϕ`k) .

It follows that the length of the arc formed by the angles is non-increasing in ∆(γ) if

Pmin ≥ Pcriticalcos(ϕmax)

cos(γ + ϕmax), (30)

where Pcritical is as stated in equation (27). The right-hand side of (30) is a strictly increasing function ofγ that diverges to ∞ as γ ↑ (π/2 − ϕmax). Therefore, there exists some γ∗ ∈ (0, π/2 − ϕmax) satisfyingequation (30) if and only if equation (30) at γ = 0 is true with the strict inequality sign, which correspondsto equation (27) in the Theorem V.5 statement. Additionally, if these two equivalent statements are true,then there exists a unique γmax ∈ (0, π/2−ϕmax) that satisfies equation (30) with the equality sign, namelyγmax = arccos(cos(ϕmax)Pcritical/Pmin)− ϕmax.

These assumptions establish the positive invariance of ∆(γ), which guarantees then together withTheorem V.1 the synchronization claimed in statement 2) of Theorem V.5.

For zero phase shifts the following corollary directly results from Theorem V.2 and Theorem V.5:

Corollary V.7 Consider the non-uniform Kuramoto-model (9) and the system of its phase differences(26), where the graph induced by P = P T is symmetric and complete. Assume that the assumptions ofTheorem V.5 hold with ϕij = 0 for all i, j. Then it follows that θ∞ ≡ Ωc and the synchronization rateis no worse than λfe stated in equation (22).

In summary, Theorem V.5 and Corollary V.7 present sufficient conditions for the synchronization ofthe non-uniform Kuramoto oscillators and are based on the bound in equation (27). Condition (27) is aworst-case (infinity) bound, both on the parameter values and on the initial geodesic distances among twoangles, which have upper-bounded by π/2. Note also that Theorem V.5 and Corollary V.7 are based onthe assumption that P = P T induces a complete graph. In simulations synchronizing behavior can beobserved for a larger set of initial conditions and when the underlying graph is not necessarily complete;we discuss these results in the next section.

In the remainder of this section we consider the case of zero phase shifts ϕij = 0 and symmetriccoupling P = P T . Under these simplifying assumptions, we can apply an ultimate boundedness argumentto relax the assumptions of Theorem V.5 and Corollary V.7, and assume only that the graph induced byP = P T is symmetric and connected and that the two-norm of the initial vector of geodesic distancesamong the angles θ is upper-bounded by π. We start our discussion with some preliminary notation andconcepts.

Given a variable x ∈ Rn and the incidence matrix H of the symmetric and connected graph induced byP = P T , recall that the vector of all difference variables is Hx = (x2 − x1, . . . ). This notation allows usto rewrite phases differences system (31) with zero phase shifts and symmetric graph induced by P = P T

in a compact vector notation as

d

d tHθ = HD−1ω −HD−1HT diag(Pij) sin(Hθ) . (31)

Note that system (31) directly reveals the underlying graph via the phase-weighted Laplacian

HT diag(Pij) sin(Hθ) =(HT diag(Pij sinc(θi − θj))H

)θ = L(Pij sinc(θi − θj))θ .

Remark V.8 (Well-posedness of the phase differences system:) The set of differential equations (31)is well defined on Tn. Adopting differential geometric language, the left-hand side of (31) is the vector

19

of frequency differences Hθ = (θ2 − θ1, . . . ) taking values in the tangent space to Tn. The vectorsin(Hθ) = (sin(θ2 − θ1), . . . ) in the right-hand side of (31) is well defined and a smooth function ofθ ∈ Tn.

With slight abuse of notation, we denote the two-norm of the vector of pairwise geodesic distancesby ‖Hθ‖2 = (

∑i,j |θi − θj|2)1/2, and aim at ultimately bounding the evolution of ‖Hθ(t)‖2. Following

a classic Kuramoto analysis [33], [29], [28], [26], we note that the non-uniform Kuramoto model (9)with ω ≡ 0 constitutes a Hamiltonian system with the Hamiltonian U(θ)|ω=0 defined in equation (6).An analysis of the ultimate boundedness of (31) by Hamiltonian arguments is possible, but results invery conservative conditions. In the recent Kuramoto literature [32], [33], a different Lyapunov functionconsidered for the uniform Kuramoto model (2) is simply ‖Hθ‖22. Unfortunately, in the case of non-uniform rates Di this function’s Lie derivative turns out to be sign-indefinite. The main reason is that theterms Pij/Di are asymmetric. However, it is possible to identify a similar Lyapunov function that has aLie derivative with symmetric coupling. Let

D6=i :=∏

k 6=iDk and D6=i,j :=

∏k 6=i,j

Dk,

and consider the function W : Tn → R defined by

W(θ) =1

2

∑i,j

1

D 6=i,j|θi − θj|2 . (32)

The analysis of the phase differences system (31) via the Lyapunov function W leads to the followingtheorem on the synchronization of non-uniform Kuramoto oscillators with zero phase shifts and initialbounded two-norm of the pairwise geodesic distances among the angles.

Theorem V.9 (Synchronization for Zero Phase Shifts) Consider the non-uniform Kuramoto model (9)and the phase differences system (31) with ϕij = 0 for all i, j ∈ 1, . . . , n2, where the graph inducedby P = P T is symmetric and connected. Assume that the algebraic connectivity is larger than a criticalvalue,

λ2(L(Pij)) > λcritical :=n ‖HD−1ω‖2

κµmini,jD6=i,j, (33)

where κ :=∑n

k=1(1/D6=k) and µ :=√

mini 6=jDiDj/maxi 6=jDiDj. Accordingly, define φmin ∈(0, π/2) as the unique solution to sinc(π − φmin) = (2/π)λcritical/λ2(L(Pij)).For the non-uniform Kuramoto model,

1) phase locking: for every φ ∈ (φmin, π/2) and for every ‖Hθ(0)‖2 ≤ µ(π− φ), there is T ≥ 0 suchthat ‖Hθ(t)‖2 < π/2 for all t > T ;

2) frequency entrainment: for every ‖Hθ(0)‖2 ≤ µ(π − φmin) the frequencies θi(t) of the non-uniform Kuramoto oscillators (9) synchronize exponentially to Ωc at a rate no worse than λfe statedin equation (22); and

3) phase synchronization: if HD−1ω ≡ 0, then for every φ ∈ (0, π) and for every ‖Hθ(0)‖2 ≤µ(π−φ) the phases θi(t) of the non-uniform Kuramoto oscillators (9) synchronize exponentially ata rate no worse than

λps = −κmini,jD6=i,j sinc(π − φ)λ2(L(Pij))/n . (34)

Remark V.10 (Physical interpretation of Theorem V.9:) Condition (33) can be interpreted as follows:λ2(L(Pij)) is the algebraic connectivity of the graph coupling the oscillators, and the term κµmini,jD 6=i,j/nweights the oscillators’ different time constants Di. Note that κ ∈ O(n), and thus κµmini,jD6=i,j/nscales independently of n. The term ‖HD−1ω‖2 = ‖(ω2/D2 − ω1/D1, . . . )‖2 corresponds to the non-uniformity in the natural frequencies or the energy of the “disturbance input” entering the otherwiseexponentially stable phase difference dynamics (31). The worst-case phase synchronization rate λps can

20

be interpreted the same way, where sinc(π− φ) corresponds to the phase locking inequality ‖Hθ(0)‖2 ≤µ(π − φ).

Remark V.11 (Reduction to classic Kuramoto oscillators:) In the case of classic Kuramoto oscillators(2), the Lyapunov function (35) is equal to the one used in [32], [33]. In this case condition (33) relaxesto K > K∗critical := ‖Hω‖2, and it follows that the oscillators synchronize for every ‖Hθ(0)‖2 < π−φmin,where φmin ∈ (0, π/2) is the unique solution to the equation sinc(π − φmin) = (2/π)K∗critical/K. Thiscondition is indeed necessary to prove Theorem 4.2 in [32]. Note also that K > K∗critical = ‖Hω‖2 >‖Hω‖∞ = ωmax−ωmin resembles the bound (28) presented in Remark V.6. Similarly, in the case HD−1ω ≡0 and given φ ∈ (0, π), the phases synchronize for every ‖Hθ(0)‖2 ≤ π−φ and the synchronization rateλps in (34) equals the one given in [33, Corollary 1] for φ ↑ π/2.

For the subsequent analysis of the phase differences system (31) leading to Theorem V.9 we need todefine the difference of two angles θi and θj living on the torus. To be rigorous, we restrict our attentionto angles contained in an open half-circle:

H := θ ∈ T : θ ∈ (0, π) .

For angles in H, the difference θi−θj is defined as the angle with magnitude equal to the geodesic distance|θi − θj| and the sign such that (θi − θj) sin(θi − θj) > 0. Accordingly we denote the vector of all phasedifferences as Hθ = (θ2 − θ1, . . . ), which is consistent with the notation ‖Hθ‖2 = (

∑i,j |θi − θj|2)1/2

introduced before. Given this convention, for θ ∈ Hn, the function W defined in (32) can be written asthe scalar-valued function Hθ 7→ W (Hθ) defined by

W(θ) =1

2

∑i,j

1

D6=i,j|θi − θj|2 =

1

2(Hθ)T diag(1/D 6=i,j)(Hθ) =: W (Hθ) . (35)

For θ ∈ Hn, the derivative of W (Hθ) along trajectories of system (31) is then given by

W (Hθ) = (Hθ)T diag(1/D6=i,j)HD−1ω − (Hθ)T diag(1/D6=i,j)HD

−1HT diag(Pij) sin(Hθ).

A component-wise analysis of the right-hand side of this equality results in a “diagonal” simplificationthat we state as follows.

Lemma V.11.1 Let n ≥ 2, P = P T , and θ ∈ Hn. Then the following identity holds:

(Hθ)T diag(1/D6=i,j)HD−1HT diag(Pij) sin(Hθ) = κ (Hθ)T diag(Pij) sin(Hθ) . (36)

Proof: The left-hand side of equation (36) reads component-wise as∑i,j

1

D6=i,j(θi − θj) ·

∑k

(PikDi

sin(θi − θk + ϕik)−PjkDj

sin(θj − θk + ϕjk)

),

and the right-hand side of (36) is simply∑i,j

(∑k(1/D 6=k)

)Pij(θi − θj) sin(θi − θj) .

Recall that k ∈ 1, . . . , n and that the double index i, j can be evaluated over all phase pairs i, j ∈1, . . . , n2 since Pij = Pji = 0 if i, j is not an edge in the graph induced by P . The identityD 6=i,jDi = D6=j and the equations above motivate the shorthands

σijk := (θi − θj)(PikD6=j

sin(θi − θk) +PjkD6=i

sin(θk − θj)),

ηijk :=PijD6=k

(θi − θj) sin(θi − θj) ,

21

expressing left- and right-hand side of equation (36) when the summation symbols are left out. The identity(36) claimed in the lemma statement may now be written compactly as∑

i,j

∑n

k=1σijk =

∑i,j

∑n

k=1ηijk ,

due to the symmetries σijk = σjik, σiik = 0, ηijk = ηjik, and ηiik = 0. The identity (36) can now beverified after some index manipulations:

2∑i,j

n∑k=1

σijk =n∑i=1

n∑j=1

n∑k=1

σijk

=n∑i=1

n∑j=1

n∑k=1

(θi − θj)(Pik/D6=j) sin(θi − θk) +n∑i=1

n∑j=1

n∑k=1

(θi − θj)(Pjk/D6=i) sin(θk − θj)

=n∑i=1

n∑k=1

n∑j=1

(θi − θk)(Pij/D 6=k) sin(θi − θj) +n∑k=1

n∑j=1

n∑i=1

(θk − θj)(Pij/D6=k) sin(θi − θj)

=n∑i=1

n∑j=1

n∑k=1

(Pij/D 6=k)(θi − θj) sin(θi − θj) =n∑i=1

n∑j=1

n∑k=1

ηijk = 2∑i,j

n∑k=1

ηijk.

The following lemma will help us to upper-bound the derivative W (Hθ) by the algebraic connectivityof the graph induced by P = P T .

Lemma V.11.2 Consider a graph with n nodes induced by P = P T with incidence matrix H andLaplacian L(Pij). It holds for any x ∈ Rn that

(Hx)T diag(Pij)(Hx) ≥ λ2(L(Pij))

n‖Hx‖22 .

Proof: Given a vector x ∈ Rn, we let x⊥ = (In− 1n11T )x be the the component of x that is projected

on the subspace orthogonal to 1. If we let Hc be the incidence matrix of the complete graph with n nodes,then (In− 1

n11T ) = 1

nHTc Hc is the Laplacian of the complete graph with uniform weights 1/n. Consider

now the following inequality and the identities

(Hx)T diag(Pij)(Hx) = xTHT diag(Pij)Hx = xTL(Pij)x

≥ λ2(L(Pij)) ‖x⊥‖22 =λ2(L(Pij))

n2

∥∥HTc Hcx

∥∥22

=λ2(L(Pij))

n2(Hcx)THcH

Tc (Hcx) ,

where we applied the Courant-Fischer Theorem. In order to continue note two things: First, HcHTc and

the complete graph’s Laplacian HTc Hc have the same eigenvalues, namely n and 0. Second, range(Hc)

and ker(HTc ) are orthogonal complements. It follows that

λ2(L(Pij))

n2(Hcx)THcH

Tc (Hcx) =

λ2(L(Pij))

n2n ‖Hcx‖22 .

Finally, note that ‖Hcx‖22 ≥ ‖Hx‖22 and the inequality claimed in the lemma follows.

Given Lemma V.11.1 and Lemma V.11.2 about the time derivative of W (Hθ), we are now in a positionto prove Theorem V.9 via standard Lyapunov and ultimate boundedness arguments.

Proof of Theorem V.9: Let φ ∈ (0, π). Assume that at some time point t it holds that ‖Hθ(t)‖∞ ≤‖H(θ(t)‖2 ≤ π − φ. In the following, assume that a coordinate translation θ(t) 7→ θ(t) + c(t) is appliedfor some appropriate c : R→ [0, π] such that θ(t) ∈ Hn. Note that such a coordinate translation does not

22

change the dynamics (31). Because ‖Hθ‖∞ ≤ π − φ, it follows that 1 ≥ sinc(θi − θj) ≥ sinc(π − φ). Inview of this inequality, of Lemma V.11.1, and of Lemma V.11.2, the bound ‖Hθ‖2 ≤ π − φ implies that

W (Hθ) ≤ (Hθ)T diag(1/D6=i,j)HD−1ω − κ(sin(φ)/(π − φ)) (Hθ)T diag(Pij)(Hθ)

≤ (1/mini,jD6=i,j) ‖Hθ‖2∥∥HD−1ω∥∥

2− κ sinc(π − φ)

λ2(L(Pij))

n‖Hθ‖22 . (37)

Note that the right-hand side of (37) is strictly negative for

‖Hθ‖2 > αc :=n ‖HD−1ω‖2

κ sinc(π − φ) mini,jD 6=i,jλ2(L(Pij)).

Pick α ∈ (0, π − φ). If

n∥∥HD−1ω∥∥

2< κ sinc(π − φ) mini,jD 6=i,jλ2(L(Pij))α , (38)

then the ratio ν := (αc/α) takes values in (0, 1). Additionally, for all ‖Hθ‖2 ∈ [α, π − φ], the right-handside of (37) is upper-bounded by

W (Hθ) ≤ −(1− ν)κ sinc(π − φ)λ2(L(Pij))

n‖Hθ‖22 .

Note that W (Hθ) may be upper and lower bounded by constants multiplying ‖Hθ‖22 as follows:

‖Hθ‖222 maxi,jD6=i,j

≤ W (Hθ) ≤ ‖Hθ‖222 mini,jD6=i,j

. (39)

To guarantee the ultimate boundedness of Hθ, two sublevel sets of W (Hθ) have to be squeezed into theset Hθ : ‖Hθ‖2 ∈ [α, π− φ] where we know W (Hθ) is strictly negative. This is possible [47, Section4.8] if

α <√

mini,jD6=i,j/max

i,jD6=i,j(π − φ) = µ(π − φ) . (40)

Standard ultimate boundedness arguments [47, Theorem 4.18] imply that, for every ‖Hθ(0)‖2 ≤ µ(π−φ),there exists a T ≥ 0 such that ‖Hθ(t)‖2 ≤ (1/µ)α for all t ≥ T . Additionally, because W (Hθ) is strictlynegative at the boundary of ‖Hθ‖2 ≤ (1/µ)α, it follows that ‖Hθ(t)‖2 < (1/µ)α for all t > T . If wechoose α = µπ/2, then equation (40) reduces to φ < π/2 and equation (38) reduces to the condition

λ2(L(Pij)) >2n

π sinc(π − φ)

‖HD−1ω‖2κµmini,jD6=i,j

. (41)

Now, we perform a final analysis of this bound. The right-hand side of (41) is a decreasing function of φthat diverges to∞ as φ ↓ 0. Therefore, there exists some φ∗ ∈ (0, π/2) satisfying equation (41) if and onlyif equation (41) is true at φ = π/2. The latter condition is equivalent to the inequality (33). Additionally, ifthese two equivalent statements are true, then there exists a unique φmin ∈ (0, π/2) that satisfies equation(41) with the equality sign, namely sinc(π − φmin) = (2/π)λcritical/λ2(L(Pij)). This concludes the proofof the phase locking statement 1).

Statement 1) implies that there exists a T ∗ > T and γ ∈ (0, π/2) such that ‖Hθ(t)‖∞ ≤ ‖Hθ(t)‖2 ≤π/2− γ < π/2 for all t ≥ T ∗. An explicit value of γ can be obtained by setting α = µ(π/2− γ) in thederivations of statement 1) above. This proves the positive invariance assumption of Theorem V.2 and thefrequency entrainment statement 2) follows immediately.

Finally, if HD−1ω ≡ 0, then, for every ‖Hθ‖2 ≤ π−φ, the right-hand side of (37) can directly be upper-bounded by W (Hθ) ≤ −2λps W (Hθ), where λps is defined in equation (34). It follows that W (Hθ(t)) ≤W (Hθ(0))e−2λpst for every ‖Hθ(0)‖2 ≤ µ(π − φ). The factor µ appears because W (Hθ) is positivelyinvariant only in sublevel sets. Reusing the bounds (39), we arrive at ‖Hθ(t)‖2 ≤ (1/µ) ‖Hθ(0)‖2 e−λpst

for every ‖Hθ(0)‖2 ≤ µ(π − φ). Finally, recall that connectivity of the graph induced by P impliesthat Hθ = 0 if and only if θi = θj for all i, j ∈ 1, . . . , n2. This concludes the proof of the phasesynchronization statement 3).

23

VI. SIMULATION RESULTS

The conditions given in Theorem III.2 are only sufficient and the real region of attraction for asynchronous equilibrium is certainly larger. In simulations we observe that both the given estimatefor the region of attraction as well as the worst case convergence rate λ (in the lossless case) of asynchronous equilibrium are conservative measures for both the power network (3) and the non-uniformKuramoto oscillators (9). Figure 3 shows a simulation of the power network (3) with 20 generators andthe corresponding non-uniform Kuramoto model (9) under worst-case initial phase differences: here theinitial phases are spaced equally (splay state) and the initial frequencies are chosen randomly from auniform distribution over (−0.1, 0.1). The system parameters satisfy ε = 0.12s, the natural frequenciessatisfy ωmax−ωmin = 3, the weights Pij are chosen randomly from a uniform distribution over the interval(0, 1), and the phase-shifts ϕij = arctan(Gij/Bij) are also uniformly distributed over (0, arctan(1)). Eventhough the initial conditions and network parameters do not satisfy the assumptions of Theorem III.2, allfrequencies synchronize to θ∞ = −21.21 rad s−1.

The quality of the singular perturbation approximation of the power network model (3) by the non-uniform Kuramoto model (9) seems to be independent of the network size and remains to be valid evenfor large values of ε. However, we note that the approximation becomes worse if the initial system stateis near the stability margin – a property that is obvious in Tikhonov’s theorem where ε is dependenton the initial condition of the full system. Conversely, if the network parameters and the initial statesatisfy the sufficient conditions for synchronization of the non-uniform Kuramoto oscillators, then thesingular perturbation approximation is expected to be valid also for large values of ε. Figure 4 showsa such a simulation, where ε = 0.6s is large. Initially, all oscillators are clustered with exception ofthe red oscillators, ϕij is uniformly distributed over (0, arctan(0.1)), and all other simulation parametersare as before. The conditions of Theorem V.5 are satisfied and we observe phase locking and frequencyentrainment, which is especially obvious in the case of the red oscillators. Note that since ε is large, thedamping of the power network is poor and the synchronization behavior of the generators is oscillatory,whereas the non-uniform Kuramoto oscillators synchronize like overdamped pendula. Nevertheless, afterthis initial transient in the boundary layer the singular perturbation approximation is accurate even thoughε is large, and all oscillators synchronize to θ∞ = 0.60 rad s−1.

For network parameters and initial conditions that do not fulfill the conditions of Theorem V.5 orTheorem V.9 the analytically derived stability margins are not exact, but they point out the following

t = 0.00s t = 0.03s t = 0.08s

t = 0.14s t = 0.24s t = 0.32s

t = 0.41s t = 0.51s t = 3.00s

Fig. 3. Simulation of the power network model (3) (outer circle) and the non-uniform Kuramoto model (9) (inner circle) from worst-caseinitial phase differences. Here the assumptions of Theorem III.2 are not satisfied.

24

t = 0.00s t = 0.11s t = 0.23s

t = 0.34s t = 0.46s t = 0.58s

t = 0.69s t = 0.81s t = 3.00s

Fig. 4. Simulation of the power network model (3) (outer circle), and the non-uniform Kuramoto model (9) (inner circle) under realisticinitial conditions for large ε. Here the network parameters and initial condition satisfy the conditions of Theorem III.2.

qualitative condition: if the initial phase differences, the network’s non-uniformity and its losses aresufficiently small compared to the network coupling, then we observe synchronization in simulations.Conversely, for weak coupling various interesting instability phenomena can be observed.

VII. CONCLUSIONS AND OUTLOOK

This paper studied the synchronization and transient stability problem for a power network. A novelnetwork-based approach leads to purely algebraic conditions and convergence rates, under which a network-reduction power system model is transiently stable depending on network parameters and initial phasedifferences. Our technical approach is based on the assumption that each generator is highly overdampeddue to local excitation control. The resulting singular perturbation analysis leads to the successful marriageof transient stability in power networks, Kuramoto oscillators, and consensus protocols. The translationbetween the first two apparently disjoint areas was achieved via a singular perturbation analysis. As aresult, the transient stability analysis of a power network model reduces to the synchronization analysisof non-uniform Kuramoto oscillators. The study of generalized coupled oscillator models is an interestingmathematical problem in its own right and was tackled by combining and extending different techniquesfrom all three mentioned areas.

The relationship between power networks, Kuramoto oscillators, and consensus algorithms gives rise tovarious research directions at the interface between these areas. In the field of transient stability analysis thepresented network-based results offer easily checkable conditions and a new perspective on the transientstability problem. Nevertheless, we are aware that the derived conditions are not competitive with thesophisticated numerical algorithms developed by the power systems community. One major reason for theconservatism of our results is that the considered network-reduction model features all-to-all coupling.For such an all-to-all coupled system the tools from distributed control do not achieve their full strengthand deliver conservative results compared to a direct analysis of the full system. Ongoing work addressesnetwork-preserving (non-reduced) power system models, network scalability problems, and the need formore accurate characterization of the region of attraction.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge Prof. Yoshihiko Susuki and Prof. Petar Kokotovic for their insightfulcomments that improved the presentation of this article.

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