Synchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators

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1 1 Synchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators Florian Dörfler, Member, IEEE, and Francesco Bullo, Fellow, IEEE Abstract arxiv: v2 [math.oc] 29 Jan 2010 Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization 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 coupled oscillators. Assuming overdamped generators possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, nonuniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and nonuniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and 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 largest and most complex machine engineered by humankind. Obviously, there are various instabilities arising in such a large-scale power grid, which can be classified by their physical nature, the size of the uncertainty or disturbance causing the instability, or depending on the devices, processes, and the time necessary to determine 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 solar and wind power. Since these renewable power sources are highly stochastic, there will be an increasing number of transient disturbances acting on a power grid that is expected to be even more complex and decentralized than the current one. Thus the detection and rejection of such instabilities will be one of the 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 power system to remain in synchronism when subjected to large transient disturbances. These disturbances may include faults on transmission elements or loss of load, loss of generation, or loss of system components such as transformers or transmission lines. For example, a recent major blackout in Italy in 2003 was caused by tripping of a tie-line and resulted in a cascade of events leading to the loss of synchronism of the Italian power grid with the rest of Europe [5]. The mechanism by which interconnected synchronous machines maintain synchronism is a balance of their mechanical power inputs and their electrical power outputs, which are nonlinearly dependent on the relative rotor angles among machines. In a classic setting the 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 their nominal values, and for generators subject to local excitation controllers aiming to restore synchronism. In order to analyze the stability of a synchronous operating point of a power grid and to estimate its region of attraction, various sophisticated algorithms have been developed [6], [7], especially for lossy transmission This work was supported in part by NSF grants CMS and CNS This document is a vastly revised and extended version of [1]. Florian Dörfler and Francesco Bullo are with the Center for Control, Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106, {dorfler, bullo}@engineering.ucsb.edu

2 lines with non-trivial transfer conductances [8], [9], [10], [11], [12]. Reviews and survey articles on transient stability analysis can be found in [13], [14], [15]. Unfortunately, the existing methods can cope only with physically simplified models and do not provide simple formulas to check if a power system synchronizes for a given system state and network parameters. In fact, it is outstanding problem to relate transient stability of a power network to the actual underlying network structure [16]. From a networktheoretic viewpoint, the transient stability problem is concerned with the robustness of synchronization in face 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 of autonomous agent systems. Recent surveys and monographs include [17], [18], [19]. One of the basic tasks in a multi-agent system is a consensus of the agents states to a common value. This consensus problem has 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 articles treating consensus problems the agents obey single integrator dynamics, but the synchronization of interconnected 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 system model. 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]. The synchronization 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 results and rich applications to various scientific areas is elegantly reviewed in [30], [31]. Recent works in the control community [20], [21], [32], [33] investigate the close relationship between Kuramoto oscillators and consensus networks. The three areas of power network synchronization, Kuramoto oscillators, and consensus protocols are apparently closely related, but the gap between the first and the second two topics has not been bridged yet. Indeed, the similarity between the Kuramoto oscillators model and the power network models used in transient stability analysis is striking. Even though power networks have often been referred to as systems of coupled oscillators, the similarity to a second-order Kuramoto-type model has been mentioned only very recently in the power networks community in [34], [35], where only qualitative simulation studies are carried out. In the coupled-oscillators literature, second-order Kuramoto models similar to power network models 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 as possible application [36]. In short, neither the Kuramoto nor the power systems literature has recognized and thoroughly analyzed this similarity. There are three main contributions in the present paper. As a first contribution, we present a coupledoscillator approach to the problem of synchronization and transient stability in power networks. Via a singular perturbation analysis, we show that the transient stability analysis for the classic swing equations with overdamped generators reduces, on a long time-scale, to the problem of synchronizing non-uniform Kuramoto oscillators with multiple time constants, non-homogeneous coupling, and non-uniform phaseshifts. This reduction to a non-uniform Kuramoto model is arguably the missing link connecting transient stability 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 synchronization and transient stability of a power network. To the best of our knowledge these conditions are the first ones to relate synchronization and performance of a power network directly to the underlying network parameters and initial state. Our conditions are based on different and possibly less restrictive assumptions than those obtained by classic analysis methods [6], [7], [8], [9], [10], [11], [12]. As mathematical model of a power network we consider a network-reduction model of arbitrary size. In comparison to classic analysis methods, we do not make any of the common or classic assumptions: we do not require uniform mechanical 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 2

3 3 be sufficiently small or even negligible. On the other hand, our results are based on the assumption that each generator is strongly overdamped, possibly due to internal excitation control. This assumption allows us to perform a singular perturbation analysis and study a reduced-dimension system. In simulations studies, our synchronization conditions appear to hold even if generators are not overdamped, and in the application to real power networks the approximation via the dimension-reduced system has been used successfully in academia and industrial practice. Our synchronization conditions are based on an analytic network-based approach whereas classic analysis methods [6], [7], [8], [9], [10], [11], [12] rely on numerical procedures to approximate the region of attraction of a synchronous equilibrium. Compared to classic analysis methods our analysis does not aim at providing best estimates of the region of attraction or the critical clearing time by numerical computation of stable manifolds of unstable equilibria and level sets of energy functions. Rather, we approach the outstanding problem [16] of relating synchronization and transient stability to the underlying network structure. For this problem, we derive sufficient and purely algebraic conditions. Our sufficient conditions can 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 an interesting mathematical problem in its own right. Our analysis extends the results obtained for classic Kuramoto 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 algebraic conditions on the network parameters and the system state establish the phase locking, frequency entrainment, and phase synchronization of the non-uniform Kuramoto oscillators with and without phase shifts. We emphasize that our results hold not only for non-uniform network parameters but also in the case 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 conditions in the literature on Kuramoto oscillators [32], [33], [25], [37]. In the end, these conditions guaranteeing synchronization of the non-uniform Kuramoto oscillators also suffice for the transient stability of the power network. Paper Organization: This article is organized as follows. The remainder of this section introduces some notation, recalls preliminaries on algebraic graph theory, and reviews the consensus protocol and the 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 nonuniform Kuramoto model whose synchronization analysis is presented in Section V. Section VI provides simulation studies to illustrate the analytical results. Finally, some conclusions are drawn in Section VII. Preliminaries and Notation: Given an n-tuple x 1,..., x n ), diagx i ) R n n is the associated diagonal matrix, x R n is the associated column vector, and x max and x min are the maximum and minimum elements, and x 2 and x are the 2- and -norm. The vectors 1 and 0 are the vectors of 1 s and 0 s of appropriate dimension. Given two non-zero vectors x R n and y R n, the angle between them, denoted by x, y) [0, π/2], satisfies cos x, y)) = x T y/ x y ). Given a two dimensional array {A ij } with i {1,..., n} and j {1,..., m}, we let A R n m denote the associated matrix and we define A max = max i,j {A ij } A min = min i,j {A ij }. Given a total order relation among the elements A ij, we let diaga ij ) denote the corresponding diagonal matrix. For a differentiable function f : R n R, let f be the column vector of partial derivatives f/ x i. To avoid clutter, the upper index n in summation symbols n i=1 and product symbols n i=1 is avoided whenever it is clear from the context. The set T 1 = π, π] is the torus and the product set T n is the n-dimensional torus. Given two angles θ 1 T 1 and θ 2 T 1 we define their distance θ 1 θ 2, with slight abuse of notation, to be the geodesic distance on T 1, as the reader would intuitively assume. Finally, we define the multivariable sine sin : T n [0, 1] n by sinx) = sinx 1 ),..., sinx n )) and the sinc funtion sinc : R R by sincx) = sinx)/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 R n n is the adjacency matrix. The entries of A satisfy

4 4 a ij > 0 for each directed edge i, j) E and are zero otherwise. Any nonnegative matrix A induces a weighted directed graph G. The Laplacian of G is the matrix LG) = La ij ) := diag j a ij) A R n n. The set {i, j} refers to the pair of nodes connected by either i, j) or j, i). In the following, we assume that A = A T, that is, G is topologically undirected and symmetrically weighted. In this case, the graph G is fully described by the elements a ij with i j. If a number k nn 1)/2 and a weight w k = a ij is assigned to any of these edges i, j) with i > j, then the incidence matrix H R nn 1)/2) n is defined component-wise as H kl = 1 if node l is the sink node of edge k and as H kl = 1 if node l is the source node of edge k; all other elements are zero. The Laplacian equals then the symmetric matrix LG) = H T diagw k )H. If G is connected, then kerh) = kerlg)) = span1), and all n 1 remaining non-zero eigenvalues of LG) are strictly positive. The Laplacian s second-smallest eigenvalue λ 2 LG)) is called the algebraic connectivity of G and, for a complete and uniformly weighted graph a ij 1 for all i and j), it satisfies λ 2 LG)) = n. Review of the Consensus Protocol and the Kuramoto Model: In a system of n autonomous agents, each characterized by a state variable x i R, one of the most basic tasks is to achieve a consensus on a common state value. In other words, we aim to achieve that, for all agent pairs {i, j} the difference variable x i t) x j t) 0 as t. Given a graph with adjacency matrix A describing the interaction between agents, a simple linear continuous time algorithm to achieve consensus on the agents state is the consensus protocol ẋ i = j a ijx i x j ), i {1,..., n}. 1) In vector notation the consensus protocol 1) takes the form ẋ = La ij )x = LG)x, which directly reveals 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 Kuramoto model, which considers n coupled oscillators with state θ i T 1 with the dynamics θ i = ω i K n sinθ i θ j ), i {1,..., n}, 2) j i where K is the coupling strength and ω i 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 Network In a power network with n generators we associate with each generator i its internal voltage level E i > 0, its self-conductance G ii > 0, and its active power output P e,i as electrical parameters. Its mechanical parameters are its mechanical power input P m,i > 0 from the prime mover, its inertia constant 1 M i > 0, its damping constant D i > 0, and its rotor angle θ i measured with respect to a synchronously rotating reference frame with frequency f 0. All these parameters are given in per unit system, except for M i and D i which are given in seconds, and the synchronous frequency f 0 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-voltage behind reactance model of interconnected swing equations [13], [38], [39]: M i πf 0 θi = P m,i E 2 i G ii D i θi P e,i, i {1,..., n}. Under the common assumption that the loads are modeled as passive admittances, all passive nodes of a power network can be eliminated Kron reduction) resulting in the symmetric reduced admittance matrix Y C n n. Its off-diagonal components are Y ij = G ij + 1 B ij for i j, where G ij 0 and B ij > 0 1 In the power system literature the standard variable for the normalized inertia constant is H i. We refrain from this notation since the variable H also represents the incidence matrix of a graph.

5 5 are the conductance between generator i and j, respectively the susceptance, in per unit values. With the highly nonlinear power-angle relationship, the electrical output power is then P e,i = j i E ie j G ij cosθ i θ j ) + B ij sinθ i θ j )). Given the admittance Y ij = Y ji between generator i and j, define the magnitude Y ij = G 2 ij + B2 ij > 0 and the phase shift ϕ ij = arctang ij /B ij ) [0, π/2). Recall that energy is lost between two generators whenever the corresponding phase shift is strictly positive and that a lossless power network is characterized by zero phase shifts between all generator pairs. The power network model can then be formulated compactly as M i πf 0 θi = P m,i E 2 i G ii D i θi j i E ie j Y ij sinθ i θ j + ϕ ij ), i {1,..., n}. 3) Typically, a dynamical model for the internal voltage of generator i is given as Ė i = ĖiE i, u i, θ i θ j ), where u i is the field excitation and can be used as a control input to the generator [40]. Note that higher order electrical and flux dynamics in damper windings can be reduced accurately into an augmented damping term D i in 3) via integral manifolds [41]. The generator s internal excitation control via a power system stabilizer essentially increases the damping torque towards the net frequency and can also be reduced into the damping term D i [38], [39]. It is commonly agreed that the classic model 3) captures the power system dynamics sufficiently well during the first swing [13], [14], [38], [39]. Thus we omit higher 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 E i = E i t) is a smooth, bounded, and strictly positive time-varying parameter. Concluding this section, the power system model 3) is also referred to as a network-reduction model because the power network is reduced to its active nodes. This renders the mathematical model tractable but structural information about the topology of the original physical power network is almost inherently lost. Alternatively, network-preserving models can be considered that are systems of differential algebraic equations. These models preserve the original network topology and are subject of the authors ongoing research. B. Synchronization and Equilibrium in Power Networks A frequency equilibrium of 3) is characterized by θ = 0 and by the power flow equations Q i θ) = P m,i E 2 i G ii P e,i 0, i {1,..., n}. 4) Clearly, system 3) cannot be in frequency equilibrium if there is an unbalance of the mechanical input power 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 θ i θ j are constant, respectively the frequency differences θ i θ j are zero. We say the power network synchronizes exponentially) if the phase differences θ i t) θ j t) become bounded and the frequency differences θ i t) θ j t) converge to zero with exponential decay rate) as t. In the literature on coupled oscillators this is also referred to as phase locking and frequency entrainment, and the case θ i = θ j for all {i, j} is referred to as phase synchronization. In order to reformulate the synchronization problem as a stability problem, system 3) is usually formulated in relative coordinates [42], for instance, in the center of angle coordinates θ i j M jθ j / j M j), or with machine n as reference in the coordinates θ i θ n, i {1,..., n 1}. Unfortunately, the resulting dynamics under these approaches are not self-contained dynamical systems. A common assumption to circumvent this obstacle is that of uniform damping, that is, D i /M i constant for all generators i.

6 6 Alternatively, sometimes the existence of an infinite bus as reference is postulated, that is, a reference generator, say n, has no dynamics, unit voltage E n 1, and zero angle θ n 0 [7], [6], [14]. Both of these assumptions are mathematical simplifications that reduce the synchronization problem to a stability analysis of the origin. We remark that in general these assumptions are not physically justified when considering the synchronization problem of generators with non-identical frequencies. C. Review of Classic Transient Stability Analysis Classically, 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) arising after 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). To answer 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 function for model 3) with non-trivial network conductance ϕ ij 0) does not exist [9], early transient stability approaches set the phase shifts ϕ ij to zero [6], [7]. In this case, the power network model 3) takes form where U : π, π] n R is the potential energy given by M/πf 0 ) θ = D θ Uθ), 5) Uθ) = i Pm,i E 2 i G ii ) θi θ i ) j i E ie j B ij cosθi θ j ) cosθ i θ j ) ). 6) As mentioned above, system 5) has to be transformed and/or simplified in order to render 5) to a self-contained system that has an equilibrium θ, 0). In this case, the energy function θ, θ) 1/2) θ T M/πf 0 ) θ+uθ) serves semi-globally as a Lyapunov function. In combination with the invariance principle, we have clearly that system 5) will converge to θ = 0 and the largest invariant zero level set of Uθ). 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 = Uθ) 7) as a dimension-reduced system. Then it is a fact that θ ) is hyperbolic equilibrium of 7) with k s stable eigenvalues if and only if θ, 0) is a hyperbolic equilibrium of 5) with k s stable eigenvalues. Moreover, the regions of attractions of both equilibria are comparable in the sense that they are bounded by the stable manifolds of the same equilibria [10, Theorem 5.7]. For further interesting relationships among the systems 5) and 7), we refer to [7], [10], [14]. Recently there have also been several developments for lossy power networks with non-trivial transfer conductances. For fixed system parameters and sufficiently small transfer conductances it is possible to analyze the power system 3) locally as a smooth perturbation of the lossless system with ϕ ij = 0 [9], [10]. Other approaches addressing lossy power networks compute numerical energy functions [8] or make use of an extended invariance principle [11], [12]. On top of the aforementioned simplifications uniform damping or infinite bus assumption), all the mentioned methods in transient stability analysis are by no means straightforward and do not result in concise and simple conditions but rather in numerical procedures approximating the stability boundaries by level sets of energy functions or stable manifolds of unstable equilibria. In the case of lossy power networks the 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 quantifies this smallness of G ij for arbitrary networks. Moreover, from a network perspective the existing algorithms say nothing about the rate of convergence, robustness, or the region of attraction with respect to the network s state, its parameters, and topology. The following sections will develop a network approach that addresses these questions quantitatively via purely algebraic tests.

7 7 III. THE NON-UNIFORM KURAMOTO MODEL AND MAIN SYNCHRONIZATION RESULT A. The Non-Uniform Kuramoto Model As 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 thoroughly analyzed 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 ω i := P m,i Ei 2 G ii effective power input to generator i) and the coupling weights P ij := E i E j Y ij maximum power transferred between generators i and j), the power network model 3) is equivalently rewritten as M i πf 0 θi = ω i D i θi j i P ij sinθ i θ j + ϕ ij ), i {1,..., n}. 8) Next, we define the non-uniform Kuramoto model by D i θi = ω i j i P ij sinθ i θ j + ϕ ij ), i {1,..., n}, 9) where we assume the parameters satisfy the following ranges: D i > 0, ω i R, P ij > 0, and ϕ ij [0, π/2), for all i, j {1,..., n}, j j; by convention, the diagonal values P ii and ϕ ii are set to zero. System 9) may be regarded as a generalization of the classic Kuramoto model 2) with multiple time-constants D i and non-homogeneous but symmetric coupling terms P ij and phase-shifts ϕ ij. The non-uniform Kuramoto model 9) will serve as a link between the power network model 3), the Kuramoto model 2), and the consensus protocol 1). Remark III.1 Second-order mechanical systems and their first-order approximations:) The non-uniform Kuramoto model 9) can be seen as an approximation of the second order system 8) for a small inertia over damping ratio M i /D i or, more specifically, for a ratio 2M i /D i much smaller than the net frequency 2πf 0. Spoken differently, system 9) may be seen a long-time approximation of 8) obtained by a singular perturbation 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 the second 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 differ only in the time constants D i on the left-hand side. The dimension-reduced system 7) is formulated as a gradient-system and is used to study the stability of the equilibria of 7), if they exist. If these equilibria do 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 underlying network structure. B. Main Synchronization Result We 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-uniform Kuramoto model 9). Theorem III.2 Main synchronization result) Consider the power network model 3) and the nonuniform Kuramoto model 9) with effective power inputs ω i := P m,i Ei 2 G ii, coupling weights P ij = E i E j Y ij, phase shifts ϕ ij, and damping terms D i. Assume that the minimal coupling weight is larger than a critical value, i.e., D max ωi P min > P critical := max ω ) ) j P lk + max sinϕ lk ). 10) n cosϕ max ) {i,j} D i D j l k D l

8 8 Accordingly, define γ max = arccoscosϕ max )P critical /P min ) ϕ max taking value in 0, π/2 ϕ max ). For γ 0, γ max ], define the non-empty) set of bounded phase differences γ) := {θ T n : max {i,j} θ i θ j π/2 γ ϕ max }. For the non-uniform Kuramoto model, 1) phase locking: for every γ 0, γ max ) the set γ) is positively invariant; and 2) frequency entrainment: if θ0) γ max ), then the frequencies θ i t) of the non-uniform Kuramoto oscillators 9) synchronize exponentially to some frequency θ [ θ min 0), θ max 0)]. For the power network model with initial conditions satisfying θ0) γ max ), 1) approximation error: for any initial frequencies θ i 0) there exists a constant ɛ > 0 such that, if the perturbation parameter ɛ := M max )/πf 0 D min ) < ɛ, then the solution θt), θt)) of 3) exists for all t 0 and it holds uniformly in t that θt) θt) = Oɛ), t 0, θt) D 1 Q θt)) = Oɛ), t > 0, where θt) is the solution to the non-uniform Kuramoto model 9) with initial condition θ0) = θ0) and D 1 Q θ) is the power flow 4) scaled by the inverse damping D 1 ; and 2) asymptotic approximation error: there exists ɛ and ϕ max sufficiently small, such that the Oɛ) approximation errors in equation 11) converge to zero as t. We discuss the assumption that the perturbation parameter ɛ needs to be small separately and in detail in 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 parameters has a direct physical interpretation when it is rewritten as n P min ωi cosϕ max ) > max ω ) j P lk + max sinϕ lk ). 12) D max {i,j} D i D j l k D l The right-hand side of 12) states the worst-case non-uniformity in natural frequencies the difference in effective power inputs at each generator) and the worst-case lossy coupling of a node to the network P lk sinϕ lk ) = E l E k G lk reflects the transfer conductance), both of which are scaled with the rates D i. These negative effects have to be dominated by np min cosϕ max )/D max on the left-hand side of 12), which P mk D m is a lower bound on min m { k cosϕ mk )}, 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 of phase locking in γ max ). In summary, the conditions of Theorem III.2 read as the network connectivity has to dominate the initial lack of phase locking, the network s non-uniformity, and the network s real losses. The weakest power flow P min in condition 10) is not only the minimal coupling weight in the network but has also a graphical interpretation as connectivity measure and is related to the physical properties of the original non-reduced) power network: if we fix the voltages E i and the phase shifts ϕ ij at constant values for all i, j {1,..., n}, then, up to some multiplicative constant, P is the all-to-all admittance matrix of the power network and 1/P min is the maximum pairwise effective resistance. The effective resistance 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 P min [44, Theorem 4.2]. Remark III.4 Refinement of Theorem III.2 in the lossless case:) Theorem III.2 can be refined in the case of a lossless network, that is, ϕ ij = 0 for all {i, j}. In this case, the worst-case conditions of Theorem III.2 on the network s parameters and its initial state can be relaxed, explicit values for the synchronization frequency and the exponential synchronization rate can be derived, and conditions for phase synchronization can be given see Theorem V.9). 11)

9 9 Remark III.5 Reduction of Theorem III.2 to classic Kuramoto oscillators:) When specialized to classic 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 later simulation 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 shows at 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 Assumption This subsection discusses the assumption that each generator is strongly overdamped which is captured by the smallness of the perturbation parameter ɛ := M max )/πf 0 D min ). This choice of the perturbation parameter ɛ and the subsequent singular perturbation analysis carried out in Section IV) is similar to what is done in the analysis of Josephson arrays [27], coupled overdamped mechanical pendula [45], and also in classic transient stability analysis in particular Theorem 5.2 in [7]). In the linear case, this singular perturbation analysis resembles the well-known example of the overdamped harmonic oscillator, which features one slow and one fast eigenvalue. In the extremely overdamped case the harmonic oscillator thus exhibits two distinct time-scales and the fast eigenvalue corresponding to the frequency damping can be neglected in the long-term phase dynamics. In the non-linear case these two distinct timescales are captured by a singular perturbation analysis with the perturbation parameter ɛ. In short, this dimension-reduction of a coupled pendula system corresponds to the physical assumption that damping and synchronization happen on separate time scales. In the application to realistic generator models one has to be careful under which operating conditions ɛ is indeed a small physical quantity. Typically, M i [2s, 8s] depending on the type of generator and the mechanical damping including damper winding torques) is poor: D i [1, 2]/2πf 0 ). However, for the synchronization problem also the generator s internal excitation control have to be considered and increase the damping torque to D i [10, 35]/2πf 0 ) depending on the system load [38], [39], [41]. In this case, ɛ O0.1) 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 network and the non-uniform Kuramoto model can also be observed for larger values of ɛ. The assumption that the quantity ɛ is small seems to be crucial for the approximation of the power network model by the non-uniform Kuramoto model. However, similar results can also be obtained independently of the magnitude of ɛ. In Remark III.1 we already discussed the similarity between the non-uniform Kuramoto model 9) and the dimension-reduced system 7) considered in classic transient stability 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 and in industrial applications [46]. Among other things, it is found that both models have the same equilibria with the same local stability properties, and the regions of attractions are also comparable [10, Theorem 5.7], as we mentioned before in Section II. These results hold independently of the magnitude of ɛ and support 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 Model In 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 10 by a singular perturbation analysis. A physically reasonable singular perturbation parameter is the worstcase choice of M i /πf 0 D i, that is, ɛ := M max. πf 0 D min The dimension of ɛ is in seconds, which makes sense since time still has to be normalized with respect to ɛ. In order to proceed consider the term F i := D i /D min )/M i /M max ), which will determine the speed of convergence of the full system 3) to the reduced system 9). With ɛ, F i, P ij, and ω i the power network model 3) can be rewritten as the singular perturbation problem ɛ θ i = F i θi + F i ω i ) D P ij sinθ i θ j + ϕ ij ), i {1,..., n}. 13) i j i Note that for ɛ = 0, system 13) reduces the non-uniform Kuramoto model 9) or, if we freeze time, it reduces to a set of algebraic equations in θ, θ) that can be solved for θ i as θ i = Q i θ)/d i, where Q i θ) is the power flow 4). For ɛ sufficiently small, the synchronization dynamics of 13) can be approximated by the non-uniform Kuramoto model 9) and the power flow 4), where the error is of order ɛ. Theorem IV.1 Singular Perturbation Approximation) Consider the power network model 3) written as the singular perturbation problem 13) with initial conditions θ0), θ0)) and solution θt, ɛ), θt, ɛ)). Consider furthermore the non-uniform Kuramoto model 9) as the reduced model with initial condition θ0) and solution θt), the quasi-steady state hθ) defined component-wise as h i θ) = Q i θ)/d i for i {1,..., n}, and the boundary layer error y i t/ɛ) := θ i 0) h i θ0)))e F it/ɛ, i {1,..., n}. Let T > 0 be arbitrary but finite and assume that the initial frequencies θ i 0) are bounded. Then, there exists ɛ > 0 such that for all ɛ < ɛ, the singular perturbation problem 13) has a unique solution on [0, T ], and for all t [0, T ] it holds uniformly in t that θt, ɛ) θt) = Oɛ), and θt, ɛ) h θt)) yt/ɛ) = Oɛ). Moreover, given any T b 0, T ), there exists ɛ ɛ such that for all t [T b, T ] and whenever ɛ < ɛ θt, ɛ) h θt)) = Oɛ). Proof: In the following we let i {1,..., n} and omit vector notation as usual in singular perturbation analysis. If we reformulate system 13) in state space with the state x, z) = θ, θ) T n R n, then we obtain the following system in singular perturbation standard form ẋ i =f i z) := z i, ɛż i =g i x, z) := F i z i + F i ω i ) D P ij sinx i x j + ϕ ij ) i j i 14). 15) The quasi-steady-state of 14)-15) is obtained by solving g i x, z) = 0 for z resulting in the unique root z i = h i x). In singular perturbation analysis, we are interested in approximating 14)-15) by the reduced system ẋ i = f i hx)) = h i x), which is equivalent to 9) in x-coordinates. The reduced system is smooth, evolves on the compact set T n, and thus its solution exists for all t 0. Let us introduce the error coordinate y i = z i h i x), which shifts the error made by the quasi-stationarity assumption z i t) = h i xt)) to the origin. After stretching time to the dimensionless variable τ = t/ɛ, the quasi-steady-state error is given by d d τ y i = g i x, y + hx)) ɛ h i f i y + hx)) = F i y i ɛ h i f i y + hx)) 16) x i x i with initial condition y i 0) = z i 0) h i x0)). By setting ɛ = 0, 16) reduces to the boundary layer model d d τ y i = F y i, y i 0) = z i 0) h i x0)). 17)

11 11 Note that 17) is globally exponentially stable and its solution depicts the boundary layer error which is equivalent y i t/ɛ) in original coordinates. The boundedness of θ0), θ0)) = x0), z0)) guarantees that y i 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 system ẋ i = f i hx)) = h i x) is bounded, and the boundary layer model 17) is exponentially stable. Thus all assumptions 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. The synchronization of system 13) happens on a slow manifold whose first order approximation is the scaled power flow θ = hθ) = D 1 Qθ). The dynamics on this slow manifold are given by the nonuniform Kuramoto model 9) and reflect the long-term synchronization behavior of system 13). The damping of system 13) happens on a fast time-scale, the boundary layer, and the initial) error of the full system frequency dynamics versus the slow quasi-stationary) synchronization manifold is given by the exponentially stable boundary layer model with convergence rate determined by D i /D min )/M i /M max ). B. Singular Perturbation Analysis on the Infinite Time Interval As usual in perturbation and averaging analysis, Theorem IV.1 holds on a finite time interval. In order to render the Oɛ) approximation valid on an infinite time interval, additionally exponential stability of the reduced system s origin is required. Among other things, the following section will show that the nonuniform Kuramoto model synchronizes exponentially under certain conditions on the network s parameters and 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 exponentially for some initial condition θ0), the singular perturbation approximation in Theorem IV.1 is valid on the infinite time interval, that is, for any T > 0. Moreover, there exist ɛ and ϕ max sufficiently small such that the approximation error converges to zero as t. Proof: The exponential synchronization of the non-uniform Kuramoto oscillators to some frequency θ i t) θ as t implies that all phase differences θ i t) θ j t) converge exponentially to constant values. After switching to a rotating frame with angular speed θ, it follows that phases θt) converge exponentially to constant values θ := lim t θt) θ t1). Consider now again the power network model in the singular perturbation standard form 14)-15), and apply the coordinate transformation x, z) 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 reduced and the boundary layer system are exponentially stable, where Lyapunov functions are readily existent by converse arguments. Therefore, the singular perturbation approximation in Theorem IV.1 can be extended to the infinite time interval 2 for any T > 0) [47, Theorem 11.2]. Now we invoke classic arguments from transient stability analysis [10] to show that the Oɛ) error converges to zero. Both the second order system 14)-15) as well as the reduced system x = fh x)) the non-uniform Kuramoto model) can be formulated with the vector of power flows Qx) as D x = Q x), 18) M/πf 0 ) ẍ = Dẋ + Qx). 19) Let the solution of 18) be termed xt), and let xt) and zt) = ẋt) be the solution of 19). By our previous argumentation, xt) converges to the origin, and xt), zt)) converge to an Oɛ) neighborhood of the origin. Due to Theorem 5.7 in [10], the origin of 19) is also locally exponentially stable for sufficiently small phase shifts ϕ ij. Thus for sufficiently small ɛ and ϕ max, the solution of 19) also converges to the origin, and the errors xt) xt) and zt) h xt)) converge exponentially to zero. 2 Note that Theorem III.2 states only a special case of the infinite time interval version of Theorem IV.1, namely the case ɛ < ɛ and t T b > 0.

12 12 V. SYNCHRONIZATION OF NON-UNIFORM KURAMOTO OSCILLATORS In this section we combine tools from transient stability analysis, consensus protocols, and Kuramoto oscillators in order to analyze the non-uniform Kuramoto model 9). We remark that system 9) can be viewed as a multiple-rate and symmetrically but non-homogeneously coupled Kuramoto system with symmetric phase shifts or as a nonlinear undirected consensus algorithm evolving on T n. Since the nonuniform Kuramoto model 9) is derived from the power network model 3), the underlying graph induced by the coupling weights P ij is complete and symmetric, i.e., except for the diagonal entries, the matrix P is fully populated all-to-all coupling) and symmetric: P ij = P ji > 0 for all {i, j} {1,..., n} 2. One difficulty in the analysis of system 9) is revealed when dividing by D i both hand sides: the weights coupling the oscillators i and j are P ij /D i on the one hand and P ij /D j on the other, that is, the coupling weights are non-symmetric. Additionally, the coupling functions themselves are also non-symmetric since the terms sinθ i θ j + ϕ ij ) and sinθ j θ i + ϕ ij ) do not add up to zero. For the sake of generality, this section considers the non-uniform Kuramoto model 9) under the assumption that the graph induced by P is neither complete nor symmetric, that is, some coupling terms P ij are zero and P P T. A. Exponential Synchronization of Phase-locked Oscillators The frequencies of the classic Kuramoto oscillators 2) synchronize only for locked phase differences, that is, the phase differences θ i t) θ j t) must be ultimately) bounded for all pairs {i, j} in order to achieve frequency entrainment [32], [33], [25], [37]. Under the assumption of locked phases, we can state an analogous result guaranteeing synchronization of the non-uniform Kuramoto oscillators 9) whenever the graph induced by P has a globally reachable node. Theorem V.1 Frequency entrainment) Consider the non-uniform Kuramoto model 9) where the graph induced by P has a globally reachable node. Assume that there exists γ 0, π/2 ϕ max ) such that the non-empty set of bounded phase differences γ) = {θ T n : max {i,j} θ i θ j π/2 γ ϕ max } is positively invariant. Then for every θ0) γ) the frequencies θ i t) of the non-uniform Kuramoto oscillators 9) synchronize exponentially to some frequency θ [ θ min 0), θ max 0)]. Proof: By differentiating the non-uniform Kuramoto model 9) we obtain the dynamical system describing the evolution of the frequencies d d t D i θ i = j i P ij cosθ i θ j + ϕ ij ) θ j θ i ). 20) Given the matrix P, consider a directed weighted graph G induced by the matrix with elements a ij = P ij /D i ) cosθ i θ j + ϕ ij ). Then the frequency dynamics 20) can be written compactly in vector form as the consensus protocol d d t θ = La ij ) θ, 21) where the matrix La ij ) is a phase-weighted Laplacian corresponding to G. By assumption we have for every θ0) γ) that θt) γ) for all t 0. Consequently the weights a ij t) = P ij /D i ) cosθ i t) θ j t) + ϕ ij ) are non-degenerate, i.e., zero for P ij = 0 and otherwise always strictly positive for all t 0. Note also that system 21) evolves on the tangent space of T n, that is, the Euclidean space R n. Therefore, the dynamics 21) can be analyzed as a linear system for the velocities θ i, where La ij ) is a state-dependent matrix, that we analyze as if it was a just time-varying Laplacian matrix. Note that the weights a ij t) are bounded continuous functions of time and induce integrated over any non-zero time interval a graph with non-degenerate weights and a globally reachable node. Furthermore, at each time instance the matrix La ij t)) is Metzler with zero row sums. Thus 21) is a linear consensus

13 13 algorithm for the velocities θ i, and it follows from the contraction property [48, Theorem 1] that θ i t) [ θ min 0), θ max 0)] for all t 0, and the dynamics 9) converge exponentially to the set, where θ i = θ j = θ for all {i, j} {1,..., n} 2. For zero phase shifts ϕ ij = 0 and symmetric coupling terms P = P T, the oscillators 9) are symmetrically coupled up to the damping terms D i appearing in the weights P ij /D i. In this case, it is possible to derive an explicit synchronization frequency and give an estimate for the exponential synchronization rate depending on the network parameters and the initial state. Towards this we borrow the center of angle formulation from transient stability analysis and introduce the constant weighted mean frequency Ω c defined as Ω c := i D i θ i i D i = i ω i Analogously to the classic Kuramoto oscillators 2) which synchronize to the average frequency 1/n) i ω i, the non-uniform Kuramoto oscillators 9) synchronize to the weighted mean frequency Ω c. 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 i D i γ) = {θ T n : max {i,j} θ i θ j π/2 γ} is positively invariant. Then for every θ0) γ) the frequencies θ i t) of the non-uniform Kuramoto oscillators 9) synchronize exponentially to the weighted mean frequency Ω c = i ω i/ i D i, and the exponential synchronization rate is no worse than. λ fe = λ 2 LP ij )) sinγ) cos D1, 1)) 2 /D max. 22) In the definition of the convergence rate λ fe in 22), the factor λ 2 LP ij )) is the algebraic connectivity of the graph induced by P = P T, the factor 1/D max is the slowest time constant of the non-uniform Kuramoto system 9), the proportionality λ fe sinγ) reflects the phase locking, and the proportionality λ fe cos D1, 1)) 2 reflects the fact that the error coordinate θ Ω c 1 is for non-uniform damping terms D i not orthogonal to the agreement vector Ω c 1. Remark V.3 Reduction of Theorem V.2 to classic Kuramoto oscillators:) Theorem V.2 and its proof are inspired by average consensus protocols [17], [18] and Theorem 3.1 in [32]. When specialized to the classic Kuramoto oscillators 2) and using the lower bound λ 2 LP ij )) np min 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 the proof of Theorem V.1. For zero shifts the consensus protocol 20) reads as d d t D i θ i = j i P ij cosθ i θ j ) θ j θ i ). 23) For each coupling term P ij 0 define the strictly positive weight w ij := P ij cosθ i θ j ) > 0 depending on the phases θt) γ). Let H be the incidence matrix of the symmetric graph induced by P = P T, and consider the symmetric phase-weighted Laplacian matrix Lw ij ) = H T diagw ij )H. This allows us to rewrite 23) compactly as a symmetric consensus protocol with multiple rates D: d d t D θ = Lw ij ) θ. 24) From Theorem V.1 it follows that the oscillators synchronize exponentially to some frequency θ. Since Lw ij ) is symmetric, it holds that 1 T d d t D θ = 0, or equivalently, i D i θ i t) is a constant conserved

14 14 quantity. If we apply this argument again at θ := lim t θi t), then we have i D i θ i t) = i D i θ, or equivalently, the frequencies synchronize exponentially to θ = Ω c. In order to derive an explicit synchronization rate to Ω c, we introduce the weighted disagreement vector δ = θ Ω c 1, as an error coordinate satisfying 1 T Dδ = 1 T D θ 1 T DΩ c 1 = i D i θ i i D i θ i = 0, that is, δ lives in the weighted disagreement eigenspace of dimension n 1 and with normal vector D 1. Since Ω c is constant and kerlw ij )) = span1) due to connectivity), the dynamics 21) written in δ-coordinates simplify to the weighted disagreement dynamics d d t Dδ = Lw ij) δ. 25) Consider the weighted disagreement function δ δ T Dδ and its derivative along the disagreement dynamics 25) which is d d t δt Dδ = 2 δ T Lw ij )δ. Note that δ is a vector in the weighted disagreement eigenspace, that is, δ T D1 = 0. Therefore, in general δ span1) and δ can be uniquely decomposed into orthogonal components as δ = 1 T δ/n) 1 + δ, where δ is the orthogonal projection of δ on the subspace orthogonal to 1. By the Courant-Fischer Theorem [49], the time derivative of the weighted disagreement function can be upper-bounded with the second-smallest eigenvalue of the Laplacian Lw ij ), i.e., the algebraic connectivity λ 2 Lw ij )), as d d t δt Dδ 2 λ 2 Lw ij )) δ 2. Since the Laplacian can be written as Lw ij ) = H T diagp ij cosθ i θ j ))H, the algebraic connectivity λ 2 Lw ij )) can be lower-bounded as λ 2 Lw ij )) λ 2 LP ij )) min {i,j} {cosθ i θ j ) : θ γ)} λ 2 LP ij )) sinγ). 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 obtained as δ = δ T 1 = δ cos δ, 1 )). The vectors δ and 1 each live on n 1)-dimensional linear hyperplanes with normal vectors D1 and 1, respectively, see Figure 1 for an illustration. The angle δ, 1 ) is upper-bounded by max δ δ, 1 ), which is said to be the dihedral angle and its sine is the gap between the two subspaces [49]. Since both hyperplanes are of co-dimension 1, we obtain the dihedral angle as the angle between the normal vectors D1 and 1, and it follows that δ, 1 ) D1, 1) with equality for n = 2). In summary, we have δ δ δ cos D1, 1)). δ δ D1, 1) D1 1 Fig. 1. Qualitative illustration of the disagreement eigenspace and the orthogonal complement of 1