N Barabanov
email: nikita.barabanov@ndsu.edu

J Schiffer
email: j.schiffer@leeds.ac.uk

R Ortega
email: ortega@lss.supelec.fr

D Efimov
email: denis.efimov@inria.fr

Conditions for Almost Global Attractivity of a Synchronous Generator Connected to an Infinite Bus

Nikita Barabanov, Johannes Schiffer, Romeo Ortega and Denis Efimov

Abstract-Conditions for existence and global attractivity of the equilibria of a realistic model of a synchronous generator with constant field current connected to an infinite bus are derived. First, necessary and sufficient conditions for existence and uniqueness of equilibrium points are provided. Then, sufficient conditions for local asymptotic stability and almost global attractivity of one of these equilibria are given. The analysis is carried out by employing a new Lyapunov-like function to establish convergence of bounded trajectories, while the latter is proven using the powerful theoretical framework of cell structures pioneered by Leonov and Noldus. The efficiency of the derived sufficient conditions is illustrated via extensive numerical experiments based on two benchmark examples taken from the literature.

Index Terms-Power system dynamics, power system stability, nonlinear control systems, cell structures.

I. INTRODUCTION

A. Motivation

Electric energy is the most fundamental energy carrier in modern industrialized societies. To satisfy the electricity demand, power systems have continuously grown to become very large, complex and highly nonlinear systems with an immense variety of actuators, controls and operational constraints [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF], [START_REF] Anderson | Power System Control and Stability[END_REF], [START_REF] Glover | Power System Analysis and Design[END_REF]. In addition, power systems are persistently being subjected to disturbances, such as changes in load, outages of power plants, or failures in transformer substations and power lines [START_REF] Kundur | Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions[END_REF]. Consequently, ensuring the stable, reliable and efficient operation of a power system is an enormous, yet crucial, task. This fact becomes even more obvious by noting that already relatively small local disturbances can lead to a cascade of failures, which can cause severe blackouts affecting millions of people [START_REF] Andersson | Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic performance[END_REF], [START_REF] Buldyrev | Catastrophic cascade of failures in interdependent networks[END_REF].

As a result, the problem of power system stability has attracted significant attention by academics and practitioners alike, with the first studies dating back, at least, to the 1920s [START_REF] Rüdenberg | Elektrische Schaltvorgänge und Verwandte Störungserscheinungen in Starkstromanlagen[END_REF], [START_REF]Transient Performance of Electric Power Systems[END_REF]-see [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF]Chapter 1] for a review of the research history on power system stability analysis. However, even today many basic questions remain open. A main reason for this is the complexity of the dynamics of power systems and even their individual components. This forces researchers to invoke several assumptions simplifying the mathematical tasks of stability analysis and control design. Standard assumptions comprise, e.g., neglecting fast dynamics [START_REF] Venkatasubramanian | Fast timevarying phasor analysis in the balanced three-phase large electric power system[END_REF], [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF], constant voltage amplitudes and small frequency variations [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF]Chapter 11]. Such assumptions permit to derive reducedorder synchronous generator (SG) models [1, Chapter 11] and employ algebraic line models [START_REF] Venkatasubramanian | Fast timevarying phasor analysis in the balanced three-phase large electric power system[END_REF], [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF], simplifying the analysis. Yet most of the employed assumptions are not physically justifiable in generic operation scenarios. In particular, the motion of the machine rotor, i.e., the swing equation, is commonly represented in terms of mechanical and electrical power instead of their corresponding torques. Unfortunately, this approximation is only valid for small frequency variations around the nominal frequency [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF], [START_REF] Anderson | Power System Control and Stability[END_REF], [START_REF] Kundur | Power System Stability and Control[END_REF]; see also the discussions in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF], [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], [START_REF] Caliskan | Uses and abuses of the swing equation model[END_REF].

The necessity of improved power systems stability analysis methods has become more compelling in the last few years. Indeed, driven by environmental, economic and societal factors, power systems worldwide are currently undergoing drastic changes and the reliable and efficient integration of large shares of renewable energy sources represents a major technical challenge for the operation of future power systems [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Winter | Pushing the limits: Europe's new grid: Innovative tools to combat transmission bottlenecks and reduced inertia[END_REF]. One of the main reasons for this is that many renewable energy sources are fluctuating by nature, hence representing an unknown time-varying disturbance to the network. Although there are a variety of concepts and approaches to facilitate the integration of renewable energy sources-e.g., demand-side management, microgrids, smart homes, etc. [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF]-it is expected that in the future power systems will operate more often closer to their security margins [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Winter | Pushing the limits: Europe's new grid: Innovative tools to combat transmission bottlenecks and reduced inertia[END_REF]. Therefore, the problem of power system stability can be expected to become even more relevant, making it important to derive easily and quickly verifiable analytic conditions for stability. This is the topic addressed in the present work.

B. Existing literature

As discussed above, one main limitation of current power system stability analysis methods is that they are based on reduced-order models only valid within a limited range of operating conditions. Extending these methods to models containing more accurate representations of the system's components and, hence, being valid in a broader range of operating conditions is a long-standing problem in the power systems literature. Yet only very recently, there has been some progress on providing solutions to this problem. The most relevant contributions are reviewed below.

In [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF] a port-Hamiltonian model of a multi-machine power system with SGs is derived. However, the stability analysis therein is restricted to the special-and unrealistic-case of an SG connected to a constant linear load. In [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] the model proposed in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF] is adopted and a sufficient condition for global asymptotic stability of a generic SG-based power system is derived. However, this stability claim critically relies on the construction of a unique steady-state solution of the power system. Unfortunately, this construction is not very realistic, as it is not only based on the assumption of a constant field winding current, but also, and more importantly, on the definition of a very specific value for the mechanical torque of each SG in the system. This torque depends on the constant field current, as well as on an arbitrarily chosen synchronization frequency and arbitrarily pre-specified steadystate terminal voltages for all SGs. The two latter dependencies significantly reduce the practical significance of the considered setup.

A classical, very well known, scenario in the power systems literature is the single-machine infinite bus (SMIB) scenario [START_REF] Kundur | Power System Stability and Control[END_REF], [START_REF] Anderson | Power System Control and Stability[END_REF]. Thereby, the infinite bus represents a three-phase voltage source with constant amplitude and frequency and the dynamics contained in the model stem from a single SG [START_REF] Anderson | Power System Control and Stability[END_REF]. Physically, this scenario is motivated by the assumption that the SG under consideration is connected to a very stiff power system, such that the one generator does not have any influence on the overall network behavior. The SMIB model is often used to illustrate and understand the basic functionality of an SG connected to a power system. It turns out that even for this simplified setup a rigorous analysis is very challenging. Yet, a clear understanding of the SMIB scenario is a fundamental step in the development of a thorough mathematical framework for the analysis of more complex power system dynamics under less restrictive and more realistic assumptions as in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF], [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] or in the classical literature [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF], [START_REF] Anderson | Power System Control and Stability[END_REF], [START_REF] Kundur | Power System Stability and Control[END_REF]. In addition, the insights that can be gained from the derived conditions can also provide valuable practical conclusions for the design, control and operation of SGs. The same applies to the emerging strand of control concepts that intend to operate grid-connected inverters such that they mimic the dynamics of conventional SGs [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], [START_REF] Visscher | Virtual synchronous machines for frequency stabilisation in future grids with a significant share of decentralized generation[END_REF], [START_REF] Beck | Virtual synchronous machine[END_REF], [START_REF] Jouini | Grid-friendly matching of synchronous machines by tapping into the DC storage[END_REF]. This is an interesting approach, because inverter-interfaced units are becoming more and more predominant in the course of growing renewable generation. However, if not properly accounted for, this development may have severe impacts on the network dynamics and stability [START_REF] Farhangi | The path of the smart grid[END_REF], [START_REF] Winter | Pushing the limits: Europe's new grid: Innovative tools to combat transmission bottlenecks and reduced inertia[END_REF]. In that context, the stability conditions derived in this work could be employed to derive tuning criteria for the control concepts proposed in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], [START_REF] Visscher | Virtual synchronous machines for frequency stabilisation in future grids with a significant share of decentralized generation[END_REF], [START_REF] Beck | Virtual synchronous machine[END_REF], [START_REF] Jouini | Grid-friendly matching of synchronous machines by tapping into the DC storage[END_REF]. The latter objective also has motivated the related works [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF].

Motivated by the abovementioned reasons, the present paper is dedicated to the global analysis of the SMIB model containing a detailed nonlinear SG model derived from first principles. By employing the same SG model, the SMIB scenario is also adopted in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF] where sufficient conditions for almost global asymptotic stability (GAS) are derived. The analysis in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF] proceeds along the classical lines of constructing an integro-differential equation resembling the forced pendulum equation and, subsequently, showing that the SMIB system is almost globally asymptotically stable if and only if the same holds for that equation. In [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF] the same authors provide slightly simpler conditions for stability resulting from verifying if a real-valued nonlinear map defined on a finite interval is a contraction. But, as stated in [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], these conditions are hard to verify analytically. Furthermore, and perhaps more importantly, the geometric tools employed to establish the results in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF], cannot be extended to a multi-machine power system. In [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] a scenario similar to that of the SMIB system is analyzed. However, as mentioned above, the analysis in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] is conducted under very stringent assumptions on the specific form of the infinite bus voltage, as well as the admissible values of the mechanical torque.

C. Contributions

In the present paper we focus on the problem of global attractivity of the SMIB model with a detailed SG model derived from first principles. The main contributions of the present paper are as follows.

• Provide a necessary and sufficient condition for uniqueness and existence of equilibria-an essential prerequisite for any convergence or stability analysis. It is shown in the paper that there are, at most, two equilibrium points. • Derive sufficient conditions for local asymptotic stability and almost global attractivity of one of these equilibrium points, i.e., for all initial conditions, except a set of measure zero, the solutions of the SMIB system tend to that equilibrium point. • Evaluate the effectiveness of the conditions via extensive numerical experiments based on two benchmark examples. Our results differ from the ones in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF] in the following aspects. First, to construct the error system we employ a coordinate transformation that only depends on the system's stationary electrical frequency, but is independent of the specific phase angle of any electrical network quantity. This is common practice in modeling of reduced-order phasorbased multi-machine power systems [START_REF] Anderson | Power System Control and Stability[END_REF], [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF] and has the advantage that the employed approaches are scalable in the sense that they are applicable-with a suitably modified, but structurally similar, Lyapunov-like function-to large multimachine systems. Second, we don't rely on any of the questionable assumptions made in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF].

At this point, it seems convenient to clarify two important technical issues regarding the conventions employed in the present paper compared to those in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF]. First, the equilibrium of the SMIB model considered in the paper cannot be rendered GAS via continuous feedback, hence the need for the qualifier "almost"1 . As a matter of fact, the SMIB system is naturally defined on the torus, which is not diffeomorphic to the Euclidean space, and GAS is hampered by a well known topological obstruction [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]. Second, the analysis carried out in the paper does not rely on the construction of a bona fide Lyapunov function. Consequently, it is not proven that the equilibrium is GAS, but only almost globally attractive. Furthermore, in order to establish this result, it is necessary to unfold the torus and work with the angles defined in Euclidean space. In turn, this implies that the angles are not bounded by definition and, hence, boundedness of solutions of the SMIB system has to be proven.

The present work extends our previous results in [START_REF] Barabanov | Almost global attractivity of a synchronous generator connected to an infinite bus[END_REF] in several regards. Namely, we provide physical interpretations of part of our conditions; the proofs of the main theoretical contributions are presented in more detail; we provide extensive numerical experiments illustrating the usefulness of our approach.

The remainder of the paper is structured as follows. In Section II, the SMIB model is introduced. The steady-state solutions of this model are investigated in Section III. Therein, it is shown that there are, at most, two equilibrium points (modulo 2π). To establish the global attractivity result, we first construct in Section IV a new Lyapunov-like function-i.e., a function that is not positive definite but whose derivative is negative semi-definite and establish convergence of bounded trajectories by invoking LaSalle's invariance principle [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]. To ensure boundedness of solutions, we recall the powerful, yet little known, theoretical framework of cell structures pioneered by Leonov and co-workers [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF], [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF], [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF] as well as Noldus [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] in Section V. Finally, in Section VI we combine the results of Sections IV and V to give conditions on the SMIB system parameters under which the cell structure principle is satisfied, hence completing the almost global attractivity analysis. Section VII presents two benchmark simulation examples illustrating the theoretical results. The paper is summarized in Section VIII where an outlook on future work is also given.

II. MODEL OF A SYNCHRONOUS GENERATOR CONNECTED TO AN INFINITE BUS

The main equations and assumptions for the considered SMIB system are given in this section. The employed assumptions on the SG are standard [START_REF] Grainger | Power System Analysis[END_REF], [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] and also used in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], [START_REF]Almost global asymptotic stability of a grid-connected synchronous generator[END_REF]. First, the rotor is round, the machine has one pole pair per phase, there are no damper windings and no saturation effects as well as no Eddy currents. Second, we assume that the rotor current i f is a real constant. Third, we assume balanced or, equivalently, symmetric three-phase signals throughout the paper [10, Definition 2.3]. For the SG this is equivalent to assuming a "perfectly built" machine connected in star with no neutral line, as in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF].

Following the notation and modeling in [START_REF] Grainger | Power System Analysis[END_REF], we employ a generator reference direction, i.e, current flowing out of the SG terminals is counted positively. The electrical rotor angle of the SG is denoted by 2 δ : R ≥0 → R and the electrical frequency by ω = δ. Here, δ is the angle between the axis of coil a of the SG and the d-axis, see Fig. 1. With a constant rotor current i f , the three-phase electromotive force (EMF) e abc : R ≥0 → R3 is given by [START_REF] Grainger | Power System Analysis[END_REF], [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]

e abc = M f i f ω   sin(δ) sin(δ -2π 3 ) sin(δ + 2π 3 )   , (1)
2 See Remark 1.

where M f ∈ R >0 is the peak mutual inductance. The threephase voltage at the infinite bus is denoted by

v abc := √ 2V   sin(δ g ) sin(δ g -2π 3 ) sin(δ g + 2π 3 )   , (2)
where V ∈ R >0 is the root-mean-square (RMS) value of the constant voltage amplitude (line-to-neutral) and

δ g = δ g (0) + ω s t ∈ R (3)
with the grid frequency ω s being a positive real constant. The stator resistance and inductance are denoted by

R ∈ R >0 , respectively L = L s + M s , where L s ∈ R >0 is the self- inductance and M s ∈ R >0
L di abc dt = -Ri abc + e abc -v abc . (4)
The mechanical equations describing the rotor dynamics, i.e., δ = ω,

J ω = -Dω + T m -T e , (5)
complete the SMIB model. Here, J ∈ R >0 is the total moment of inertia of the rotor masses, D ∈ R >0 is the damping coefficient 3 and T m ∈ R ≥0 is the mechanical torque provided by the prime mover. Furthermore, we assume T m constant throughout the paper. Finally, the electrical torque T e can be expressed as [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] T e = ω -1 i abc e abc .

For our analysis, we choose to represent all three-phase electrical variables in dq-coordinates. To this end, we employ the dqtransformation given in Appendix A with the transformation angle ϕ := ω s t.

The angle difference between the rotor angle δ and the dqtransformation angle ϕ is denoted by

θ := δ -ϕ.
Hence, the grid voltage ( 2) is given by the following constant vector in dq-coordinates (see [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF]),

v dq = v d v q = √ 3V sin(δ g -ϕ) cos(δ g -ϕ) = √ 3V sin(δ g (0)) cos(δ g (0)) ,
where the second equality follows from (3). Furthermore, with the EMF in dq-coordinates is given by

b := 3/2M f i f , (8)
a 1 a 2 + b 2 + c 1 c 2+ b 1 i f a-
e dq = e d e q = bω sin(θ) bω cos(θ) . (9)
Note that the electrical torque T e in ( 6) is given in dqcoordinates by

T e = ω -1 i dq e dq = b(i q cos(θ) + i d sin(θ)) (10)
and that, with ϕ given in [START_REF] Rüdenberg | Elektrische Schaltvorgänge und Verwandte Störungserscheinungen in Starkstromanlagen[END_REF] and T dq (•) given in Appendix A,

dT dq (ϕ) dt i abc = ω s -i q i d ,
see [10, equation (4.8)]. Hence, by replacing the rotor angle dynamics, i.e., δ, with the relative rotor angle dynamics, i.e., θ, the SMIB model given by ( 4), ( 5) and ( 6) becomes in dq-

coordinates θ = ω -ω s , J ω = -Dω + T m -b (i q cos(θ) + i d sin(θ)) , L id = -Ri d -Lω s i q + bω sin(θ) -v d , L iq = -Ri q + Lω s i d + bω cos(θ) -v q . ( 11
)
The model ( 11) is used for the analysis in this paper.

Remark 1. To establish an important result of this paper, namely that all bounded solutions converge to an equilibrium, it is necessary to-unfolding the torus where the system lives naturally-work with the angles defined on the real line rather than on the circle. This technical step is required in order to construct a continuous Lyapunov-like function for the system [START_REF] Kundur | Power System Stability and Control[END_REF]. However, the Lyapunov-like analysis does not allow us to conclude that the coordinate θ in (11) is bounded. Therefore, the latter property is established by using the framework of cell structures [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF], [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF], [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF] in Section V.

Remark 2. In [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], the dq-transformation angle ϕ is chosen as the rotor angle, i.e., ϕ = δ. This yields e d = 0 in (9). In addition, in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF] a dq-transformation is chosen in which the d-axis lags the q-axis by π/2. In our notation, this coordinate frame can be obtained by setting the transformation angle to ϕ = δ + π. Hence, e q = bω in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] and e q = -bω in [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF].

Remark 3. The analysis reported in the paper can be conducted in any coordinate frame. However, we favor the one used here since it seems to be more suitable to extend the results to the multi-machine case.

III. EXISTENCE AND UNIQUENESS OF EQUILIBRIA

This section is dedicated to deriving conditions on existence and uniqueness of equilibria of the system [START_REF] Kundur | Power System Stability and Control[END_REF], which we denote by (θ s , ω s , i s d , i s q ). To simplify the notation it is convenient to introduce two important constants

c : = b (v 2 d + v 2 q )((Lω s ) 2 + R 2 ), P : = 1 c -b 2 ω s R + (T m -Dω s )((Lω s ) 2 + R 2 ) . (12)
It is clear from ( 8) that c is nonzero if the rotor current i f is nonzero, which is satisfied in any practical scenario.

Proposition 1. The system in [START_REF] Kundur | Power System Stability and Control[END_REF] possesses two unique steady-state solutions (modulo 2π) if and only if

|P| < 1. (13)
If and only if (13) is satisfied with equality, the system (11) has exactly one steady-state solution (modulo 2π).

Proof. Obviously, the equilibria of the system (11) are 2πperiodic in θ. Furthermore, we have to solve the equations

ω -ω s = 0, -Dω -bi d sin(θ) -bi q cos(θ) + T m = 0, -Ri d -Lω s i q + bω sin(θ) -v d = 0, Lω s i d -Ri q + bω cos(θ) -v q = 0. (14)
It is straightforward to check that equilibria (θ s , ω s , i s d , i s q ) are given by

ω s = ω s , i s d = bRω s sin(θ s ) -bL(ω s ) 2 cos(θ s ) -v d R + v q Lω s (Lω s ) 2 + R 2 , i s q = (ω s ) 2 Lb sin(θ s ) + ω s bR cos(θ s ) -v d Lω s -v q R (Lω s ) 2 + R 2 , b(Lω s v q -Rv d ) sin(θ s ) -b(Lω s v d + Rv q ) cos(θ s ) = -b 2 ω s R + (T m -Dω s )((Lω s ) 2 + R 2 ). (15)
Some trigonometric manipulations of the left-hand side of the last equation yield

b(Lω s v q -Rv d ) sin(θ s ) -b(Lω s v d + Rv q ) cos(θ s ) = = b (v 2 d + v 2 q )((Lω s ) 2 + R 2 ) sin(θ s -φ), where φ = arctan Lω s v d + Rv q Lω s v q -Rv d , (16)
and arctan(•) denotes the standard arctangent function with range in the interval [0, π). Note that, by definition of v d and v q , the case v d = v q = 0 is excluded and, hence, φ is welldefined.

With these considerations, and by rearranging terms, the last equation in [START_REF] Farhangi | The path of the smart grid[END_REF] implies that solutions θ s exist if and only if

-b 2 ω s R + (T m -Dω s )((Lω s ) 2 + R 2 ) b (v 2 d + v 2 q )((Lω s ) 2 + R 2 ) ≤ 1, (17)
which is precisely condition [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF]. Thus, condition ( 17) is necessary and sufficient for system [START_REF] Kundur | Power System Stability and Control[END_REF] to have either one (equality) or exactly two (strict inequality) equilibria (modulo 2π), completing the proof.

A related analysis of existence and uniqueness of equilibria of the model ( 11) is reported in [13, Section III.E]. Yet, therein the admissible mechanical torque T m is restricted to a specific function T m = T m (b, D, ω s , L, R, v d , v q ). This parametric restriction is needed to ensure existence of exactly one single equilibrium point of the system [START_REF] Kundur | Power System Stability and Control[END_REF] and is essential for the subsequent GAS analysis in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF]. Compared to this, the present analysis merely requires [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] to be satisfied, hence being significantly less restrictive and practically more realistic. Also, we emphasize that the topological properties of the model [START_REF] Kundur | Power System Stability and Control[END_REF] prevent any of its equilibria being GAS [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]. The GAS claim in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] can only be established, because the dynamics of the angle θ in [START_REF] Kundur | Power System Stability and Control[END_REF] are omitted from the analysis.

In [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF], [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], no explicit analysis of existence and uniqueness of equilibria is performed, but it is claimed that for a sufficiently large infinite bus amplitude V there always are two equilibria. This claim can be made rigorous by inspection of [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], which clearly can be satisfied for sufficiently large

V = 1/3 v 2 d + v 2 q .
To provide a physical interpretation of condition [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF], it is convenient to introduce some constants following standard definitions in power systems. More precisely, we define the reactance X, the conductance G, the susceptance B and the admittance Y as

X := ω s L, G := R R 2 + X 2 , B := - X R 2 + X 2 , Y := G + jB, |Y | = G 2 + B 2 , (18)
Moreover, we define the admittance angle as

β := arctan G B , (19)
and denote the amplitude of the EMF e abc in (1) by

E := 1 √ 2 ωmi f = 1 √ 3 ωb. ( 20
)
We have the following relation between the electrical torque T e in [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF] and the active power P, i.e., the amount of power transformed from mechanical power to electrical power with some of it being lost in the resistance of the stator coils [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF],

P = ωT e .
Multiplying condition (13) on both sides with |ω s c| yields

|Pcω s | < |cω s |. (21)
By using the relations introduced above, we obtain

Pcω s = -(bω s ) 2 R + (T m -Dω s )ω s (X 2 + R 2 ) = |Y | -2 -3E 2 G + P m -P D , (22)
where we introduced the steady-state mechanical power P m = T m ω s and the steady-state damping power P D = D(ω s ) 2 . Also,

cω s = ω s bV √ 3|Y | -1 = 3EV |Y | -1 . (23)
Hence, with ( 22) and ( 23), [START_REF] Natarajan | Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus[END_REF] becomes

| -3E 2 G + P m -P D | < 3|(EV |Y |)|. ( 24
)
We recall that the stationary power flow P L over the power line connecting the SG and the infinite bus is given by

P L = 3EV |Y | sin(θ s -δ g (0) + β), ( 25
)
where β is defined in [START_REF] Beck | Virtual synchronous machine[END_REF]. Consequently, we see from [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] that condition [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] has the physical interpretation that the mechanical power provided by the machine must not exceed the power dissipated through damping (P D ), internal losses (3E 2 G) and transmission (P L ), i.e., for an equilibrium to exist a stationary power balance must be possible. A similar observation has been made for first-order droop-controlled inverters in [START_REF] Simpson-Porco | Synchronization and power sharing for droop-controlled inverters in islanded microgrids[END_REF].

IV. CONVERGENCE OF Bounded SOLUTIONS

In this section, a sufficient condition under which all bounded solutions of the system (28) converge to an equilibrium is derived. The claim is established by constructing a Lyapunov-like function and invoking LaSalle's invariance principle [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF].

Throughout the rest of the paper we make the following natural assumption.

Assumption 1. The parameters of the system (11) are such that condition (13) of Proposition 1 is satisfied and i f > 0.

As seen from ( 8), the signs of c and, hence, P defined in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF] depend on the sign of the constant rotor current i f . For the subsequent analysis, it is important to know the sign of c, as it determines which of the two equilibria of the system (11) is stable. Assumption 1 is made without loss of generality because, as seen below, the analysis for i f < 0 follows verbatim-see the numerical example in Section VII. Proposition 2. Consider the system (11) verifying Assumption 1 and the inequality

4RD[(Lω s ) 2 + R 2 ] > (Lbω s ) 2 . ( 26
)
Every bounded solution tends to an equilibrium point.

Proof. Assumption 1 ensures the existence of equilibria. As usual, it is convenient to first shift one of the (infinitely many) equilibrium points to the origin via the the change of coordinates θ = θ + θ s , ω = ω + ω s , i d = ĩd + i s d , i q = ĩq + i s q .

In the incremental variables ( θ, ω, ĩd , ĩq ) the system (11) has the form θ =ω,

J ω = -D(ω + ω s ) -b(i s d + ĩd ) sin(θ s + θ) -b(i s q + ĩq ) cos(θ s + θ) + T m , L id = -R(i s d + ĩd ) -Lω s (i s q + ĩq ) + b(ω + ω s ) sin(θ s + θ) -v d , L iq = -R(i s q + ĩq ) + Lω s (i s d + ĩd ) + b(ω + ω s ) cos(θ s + θ) -v q . ( 27
)
Furthermore, taking into account equations [START_REF] Farhangi | The path of the smart grid[END_REF], we get θ =ω,

J ω = -D ω -b ĩd sin(θ s + θ)-bi s d (sin(θ s + θ) -sin(θ s )) -b ĩq cos(θ s + θ) -bi s q (cos(θ s + θ) -cos(θ s )), L id = -R ĩd -Lω s ĩq + bω s (sin(θ s + θ) -sin(θ s )) + bω sin(θ s + θ), L iq = -R ĩq + Lω s ĩd + bω s (cos(θ s + θ) -cos(θ s )) + bω cos(θ s + θ). (28)
The second step is to construct the Lyapunov-like function.

To this end, note that the electrical dynamics takes the form

L idq = u w + bω sin(θ s + θ) bω cos(θ s + θ) , (29)
where we defined

u w := -R -Lω s Lω s -R ĩdq + bω s (sin( θ + θ s ) -sin(θ s )) bω s (cos( θ + θ s ) -cos(θ s )) .
From the expression above we see that including a term in L 2 (u 2 +w 2 ) in the Lyapunov-like function candidate generates a "good" term -R| ĩdq | 2 in its derivative. This together with a detailed inspection of (28) suggests the following function

V (χ) = L 2 (u 2 + w 2 ) + J ω2 2 (Lω s ) 2 + R 2 + b 2 Rω s ( θ -sin θ) + L(bω s ) 2 (1 -cos θ) + b (Lω s ) 2 +R 2 i s d cos θ s -cos( θ+θ s )-θ sin θ s + i s q sin( θ + θ s ) -sin θ s -θ cos θ s , (30)
where we defined the four-dimensional state vector χ := ( θ, ω, ĩd , ĩq ).

Some lengthy, but straightforward, calculations show that

V = -R[u 2 + w 2 ] -D((Lω s ) 2 + R 2 )ω 2 + + ωu Lbω s cos( θ + θ s ) -ωw Lbω s sin( θ + θ s ) = u w ω M u w ω , (31)
where we defined the matrix

M :=    -R 0 Lbω s cos( θ+θ s ) 2 0 -R -Lbω s sin( θ+θ s ) 2 Lbω s cos( θ+θ s ) 2 -Lbω s sin( θ+θ s ) 2 -D((Lω s ) 2 + R 2 )    .
(32) A simple Schur complement analysis shows that M < 0 if and only if [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF] holds. Now, by LaSalle's invariance principle [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF] all bounded solutions of the system (28) converge to the largest invariant set contained in the set {χ ∈ R 4 : V = 0}. Clearly,

V = 0 ⇔ w = u = ω = 0.
Hence, θ is constant and, from ( 29), we have that ĩdq is also constant. Consequently, the set {χ ∈ R 4 : V = 0} is an equilibrium set, completing the proof.

Remark 4. The function V given in (30) contains a linear term in θ, which makes V discontinuous if we define the system on the torus. See Remark 1.

Remark 5. With [START_REF] Visscher | Virtual synchronous machines for frequency stabilisation in future grids with a significant share of decentralized generation[END_REF], condition (26) can be rewritten as

4DG(R 2 + X 2 ) 2 > X 2 b 2 ⇔ 4D R X > |B|b 2 .
This shows that a high damping factor D and a high R/X ratio, i.e., a high electrical dissipation, are beneficial to ensure convergence. On the contrary, a high value of |b|, i.e., a high excitation and consequently large EMF amplitude, deteriorate the likelihood of convergence. Both observations are sensible from a physical perspective and consistent with practical experience.

V. BOUNDEDNESS OF SOLUTIONS

In this section, we establish a key requirement to invoke the convergence result of Proposition 2, namely boundedness of solutions. As indicated in Remark 4 to ensure continuity of the function V, we are viewing the system evolving in R 4 . Therefore θ is not a-priori bounded. To prove this fact we use the cell structure principle of Leonov and co-workers [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF], [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF], [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF] as well as Noldus [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF]. Although the proof of the proposition is an immediate corollary of Theorem 16 in [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF]Chapter 8], (see also [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF]), it is given here for the sake of completeness. Proposition 3. Consider the function V defined in [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF]. Assume there exist positive numbers and λ such that along the solutions of the system (28) the function

V := V -2 θ2 ( 33
)
verifies V ≤ -λ V . ( 34
)
Then, all solutions ( θ, ω, ĩd , ĩq ) of the system (28) are bounded.

Proof. Note that from [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF] and [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF] we have that the evolution of u, w and ω-along solutions of the system (28)-is bounded. This implies that ĩdq is also bounded and it only remains to show that θ is bounded. To show the latter, we begin by simplifying the function V defined in [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF]. Recall from ( 16) that by definition, φ is the unique number in [0, π) such that

cos φ = b c (Lω s v q -Rv d ), sin φ = b c (Lω s v d + Rv q ), (35)
where the constant c is defined in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF]. Together with [START_REF] Farhangi | The path of the smart grid[END_REF], we have that

b 2 Rω s ( θ -sin θ) + L(bω s ) 2 (1 -cos θ)+ + b (Lω s ) 2 + R 2 i s d cos θ s -cos( θ + θ s ) -θ sin θ s + + i s q sin( θ + θ s ) -sin θ s -θ cos θ s = c θ 0 [sin(θ s -φ + s) -sin(θ s -φ)]ds.
Hence, V can be written compactly as

V (χ) = L 2 (u 2 + w 2 ) + J ω2 2 ((Lω s ) 2 + R 2 )+ + c θ 0 [sin(θ s -φ + s) -sin(θ s -φ)]ds. (36)
From the definition of V in [START_REF] Teodorescu | Grid Converters for Photovoltaic and Wind Power Systems[END_REF] it follows that the function V is positive definite on the hyperplane θ = 0. For every integer k = 0, ±1, ±2, . . . consider the function

Vk (χ) = L 2 (u 2 + w 2 ) + J ω2 2 ((Lω s ) 2 + R 2 ) -2 ( θ -2πk) 2 + c θ 2πk
[sin(θ sφ + s)sin(θ sφ)]ds.

(37)

It follows immediately that Vk is positive definite on the hyperplane θ = 2πk. In addition, by denoting

χ = χ -2πk[1 0 0 0] ,
we have from (37) that for any integer k,

V ( χ) = Vk (χ).
Furthermore, by writing [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF] as

χ = f (χ),
we see that the periodicity of the dynamics [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF] in

θ implies that χ = f (χ) = f ( χ).
Hence, evaluating the time derivative of Vk along solutions of the system (28) yields

Vk = ∂ Vk (χ) ∂χ f (χ) = ∂ Vk (χ) ∂χ f (χ -2πk[1 0 0 0] ) = ∂ V (χ -2πk[1 0 0 0] ) ∂χ f ( χ).
Since ∂ χ = ∂χ and V ≤ -λ V by assumption, we obtain

Vk = ∂ V ( χ) ∂ χ f ( χ) ≤ -λ V ( χ) = -λ Vk (χ).
Hence, the fact that by assumption V ≤ -λ V , i.e., condition [START_REF] Monzón | Local and global aspects of almost global stability[END_REF], implies that also for every integer k, Vk ≤ -λ Vk .

This together with the 2π-periodicity of the system (28) with respect to θ, implies that for every integer k the set

Z k = {χ ∈ R 4 : Vk (χ) ≤ 0} (38)
is invariant with respect to solutions of the system [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF]. Assume χ(•) is a solution of system [START_REF] Gelig | Stability of nonlinear systems with nonunique equilibrium position[END_REF] with initial condition χ(0) = χ 0 . From the definition of the function Vk in (37) we see that Vk (χ 0 ) is decreasing with respect to |k| quadratically. Hence, for any χ 0 there exist integers k 1 and k 2 , with k 1 < k 2 , such that Vk1 (χ 0 ) ≤ 0, θ(0) ≥ 2πk 1 , and Vk2 (χ 0 ) ≤ 0, θ(0) ≤ 2πk 2 . The function Vk1 is positive on the plane θ = 2πk 1 , and the function Vk2 is positive on the plane θ = 2πk 2 . Furthermore, the sets Z k1 and Z k2 are invariant.

Consequently, we have that

2πk 1 ≤ θ(t) ≤ 2πk 2
for all t ≥ 0. This completes the proof. Remark 6. The cell structure method is applicable to general, non-autonomous, nonlinear systems of the form ẋ = f (x, t), x ∈ R n with the only requirement that f is periodic with respect to (a part of) the state vector x. For further details see [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF]Chapter 8].

VI. MAIN RESULT: ALMOST GLOBAL ATTRACTIVITY

To streamline the presentation of our main result, it is convenient to introduce the following functions and quantities. Given the equilibrium values θ s and ω s , define the functions

q( θ) := c θ 0 [sin(θ s -φ + s) -sin(θ s -φ)] ds, g(λ) := 4 R - Lλ 2 ((Lω s ) 2 + R 2 )(D - Jλ 2 ) - 2 min λ ,
with c and φ defined in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF] and [START_REF] Fitzgerald | Electric Machinery[END_REF], respectively, and where the constant min is given by

min := inf ∈ R >0 | q( θ) ≤ 2 θ2 , ∀ θ ∈ R . (39)
The lemma below shows that the infimum above is indeed achievable, and provides an explicit, though conservative, upper bound ¯ for min .

Lemma 1. Consider the function

h( θ) = q( θ) - ¯ 2 θ2 . (40)
For all ¯ > c we have that h( θ) ≤ 0 for all θ ∈ R.

Proof. Straight-forward calculations yield

q( θ) = c[-cos(θ s -φ + θ) -sin(θ s -φ) θ + cos(θ s -φ)].
Hence, the function h in (40) is similar to a parabola which opens downward. The critical points of h are attained at values of θ * satisfying ∂h ∂

θ θ= θ * = c[sin(θ s -φ + θ * ) -sin(θ s -φ)] -¯ θ * = 0. ( 41
)
It follows from the mean value theorem that

sin(θ s -φ + θ * ) -sin(θ s -φ) ≤ | θ * |.
Thus, for ¯ > c, the only solution of ( 41) is θ * = 0. Furthermore,

∂ 2 h ∂ θ2 θ= θ * = c cos(θ s -φ + θ * ) -¯ .
which shows that for ¯ > c ≥ c cos(θ sφ), θ * = 0 is a maximum of h, completing the proof.

The following assumption is fundamental to establish our claim.

Assumption 2. There exists λ max > 0-a point of local maximum of the function g(λ)-such that

2R > λ max L, g(λ max ) > (Lbω s ) 2 . (42)
We are now ready to state our main result.

Theorem 1. Consider the system (11) verifying Assumptions 1 and 2. The equilibrium point (θ s , ω s , i s d , i s q ) satisfying |θ s -φ| < π 2 (modulo 2π) with φ defined in (16) is locally asymptotically stable and almost globally attractive, i.e., for all initial conditions, except a set of measure zero, the solutions of the system (11) tend to that equilibrium point.

Proof. Assumption 1 ensures, via Proposition 1, that equilibrium points exist. From the derivations above it is clear that the gist of the proof is to verify the conditions of Propositions 2 and 3.

We begin by establishing the local asymptotic stability claim. From [START_REF] Farhangi | The path of the smart grid[END_REF] and the definition of q( θ) above it follows that q(0) = 0, q (0) = 0, q (0) = c cos(θ sφ)

and sin(θ s -φ) = P, (43)
with P defined in [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF]. From Assumption 1 we have |P| < 1, therefore, the equation (43) has two roots θ s (modulo 2π) in the interval [φ, φ + 2π). If |θ s -φ| < π 2 , then q (0) > 0. This implies that the function V defined in [START_REF] Leonov | Phase synchronisation. Theory and applications[END_REF] has a local minimum at the origin. Furthermore, the parameters λ and min only enter with negative sign in g(λ). Hence, Assumption 2 implies that (42) is also satisfied for λ = min = 0 (with ( min /λ)| (0,0) := 0) which is exactly condition [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]. Thus, V ≤ 0 and the zero solution of the system ( 28) is Lyapunov

asymptotically stable (see Proposition 2). If |θ s -φ| > π 2 ,
then q (0) < 0, and the zero solution of the system ( 28) is Lyapunov unstable.

To establish almost global attractivity of the asymptotically stable equilibrium point (modulo 2π), in the following we assume that the zero solution of the system (28) is Lyapunov unstable (and therefore |θ s -φ| > π 2 ). Recall the sets Z k defined in (38) and note that every intersection of sets Z k is also invariant. The set Z k is equal to Z 0 shifted in the coordinate θ by 2πk to the right since θ s k = θ s + 2πk. Now,

Z 0 ={χ ∈ R 4 : L 2 (u 2 + w 2 ) + J ω2 2 ((Lω s ) 2 + R 2 ) + q( θ) -2 θ2 ≤ 0}. (44)
From Lemma 1 we have that this set is nonempty. Next we check the condition of Proposition 3. To this end, we evaluate [START_REF] Simpson-Porco | Synchronization and power sharing for droop-controlled inverters in islanded microgrids[END_REF]. This yields

d V dt + λ V with V defined in
d V dt + λ V = -R[u 2 + w 2 ] -D((Lω s ) 2 + R 2 )ω 2 + ωu Lbω s cos( θ + θ s )-ωw Lbω s sin( θ + θ s ) -θ ω + λ[ L 2 (u 2 + w 2 ) + J ω2 2 ((Lω s ) 2 + R 2 ) +c θ 0 [sin(θ s -φ + s) -sin(θ s -φ)]ds-2 θ2 ] ≤(-R + Lλ 2 )[u 2 + w 2 ] -((Lω s ) 2 + R 2 )(D - Jλ 2 )ω 2 + ωu Lbω s cos( θ + θ s )-ωw Lbω s sin( θ + θ s ) -θ ω - λ( -min ) 2 θ2 = u w ω θ M 1 u w ω θ , (45)
where the matrix M 1 is defined in (48). The matrix M 1 is negative definite if and only if > min and the matrix M 2 defined in (49) is negative definite. Similarly to the matrix M defined in [START_REF] Grainger | Power System Analysis[END_REF] the matrix M 2 is negative definite if and only if 2R > Lλ and

4(R- Lλ 2 )[(Lω s ) 2 +R 2 )(D- Jλ 2 )- 2 2λ( -min ) ] > (Lbω s ) 2 .
(46) We now proceed to prove the existence of the positive parameters and λ appearing in (46). The maximum of the left-hand side with respect to is attained at = 2 min . For this choice, the inequality (46) takes the form

4(R - Lλ 2 )[((Lω s ) 2 + R 2 )(D - Jλ 2 ) - 2 min λ ] > (Lbω s ) 2 .
(47) Consider the following polynomial

f (λ) := ((Lω s ) 2 + R 2 )(Dλ - Jλ 2 2 ) -2 min ,
and denote by λ 1 its smallest root, that is,

λ 1 = D -D 2 -4J min (Lω s ) 2 +R 2 J . M 1 =      -R + Lλ 2 0 Lbω s cos( θ+θ s ) 2 0 0 -R + Lλ 2 -Lbω s sin( θ+θ s ) 2 0 Lbω s cos( θ+θ s ) 2 -Lbω s sin( θ+θ s ) 2 -((Lω s ) 2 + R 2 )(D -Jλ 2 ) -2 0 0 -2 -λ( -min) 2      (48) M 2 =    -R + Lλ 2 0 Lbω s cos( θ+θ s ) 2 0 -R + Lλ 2 -Lbω s sin( θ+θ s ) 2 Lbω s cos( θ+θ s ) 2 -Lbω s sin( θ+θ s ) 2 -((Lω s ) 2 + R 2 )(D -Jλ 2 ) + 2 2λ( -min)    (49) If λ 1 < 2R L , then on the interval [λ 1 , 2R L ]
there is a unique point λ max of local maximum of the left-hand side in (47).

The derivations above, together with the definition of g(λ), prove that the inequalities (42) of Assumption 2 ensure M 1 is negative definite. Hence, condition [START_REF] Monzón | Local and global aspects of almost global stability[END_REF] is satisfied and therefore Proposition 3 implies that all solutions (θ, ω, i d , i q ) of the system (11) are bounded.

Furthermore, the terms in λ and min only appear with negative sign on the right-hand side of (47). In addition, we see from the matrices M 1 in (48) and M 2 in (49) that for λ = min = = 0, (47) reduces to [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]. Hence, condition (47) only holds if condition (26) of Proposition 2 holds. Consequently, under Assumption 2, all solutions (θ, ω, i d , i q ) of the system [START_REF] Kundur | Power System Stability and Control[END_REF] are bounded and tend to an equilibrium point.

Denote by A the Jacobian of the system (11) evaluated at an equilibrium point, i.e.,

A =     0 1 0 0 b J ζ -D J -b sin(θ s ) -D J cos(θ s ) b L ω s cos(θ s ) b L sin(θ s ) -R L -ω s -b L ω s sin(θ s ) b L cos(θ s ) ω s -R L     ,
where ζ = i s q sin(θ s )i s d cos(θ s ).

Note that with [START_REF] Farhangi | The path of the smart grid[END_REF] and ( 16), we have that

ζ = 1 R L 2 + (ω s ) 2 b(ω s ) 2 L - 1 L cos(θ s -φ) R L 2 + (ω s ) 2 .

Direct computations yield

det(A) = b cos(θ s -φ) J R 2 + (Lω s ) 2 .
Recall that at the unstable equilibrium point |θ s -φ| > π 2 . Consequently, there det(A) < 0. Since moreover A is a realvalued matrix of dimension 4, det(A) < 0 implies that A has at least one positive real eigenvalue. Hence, by invoking [34, Proposition 11] we conclude that the region of attraction of the unstable equilibrium has zero Lebesgue measure. Thus, for all initial conditions, except a set of measure zero, the solutions of the system [START_REF] Kundur | Power System Stability and Control[END_REF] tend to the stable equilibrium point. This shows that the latter is almost globally attractive and completes the proof.

Remark 7. Note that if P = 0 (and therefore |θ s -φ| = π

2 ) then min = 0, and the inequality (47) is equivalent to [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF].

Remark 8. The related analysis in [START_REF] Caliskan | Compositional transient stability analysis of multimachine power networks[END_REF] critically relies on imposing a specific value for the mechanical torque T m and on the knowledge of the stationary rotor currents i s dq . Compared to this, we don't need to impose any specific value for T m . Furthermore, our convergence conditions only depend on the steady-state value of ω s and, through (39), on the relative rotor angle θ s , but are independent of the stationary rotor currents i s dq .

VII. NUMERICAL EXAMPLES

A key question is whether the conditions imposed by the inequalities (42) of Assumption 2 are verified in a practical SMIB scenario. We investigate this issue via two numerical benchmark examples reported in the literature.

The first example is taken directly from [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF] and represents a 5kW synchronverter. Note that in the example of [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF] the rotor current i f < 0. Thus, b < 0 and c < 0, see [START_REF] Fiaz | A port-Hamiltonian approach to power network modeling and analysis[END_REF]. In our notation, this corresponds to the (potentially) stable equilibrium being shifted by π. Indeed, conditions (12), ( 13) and ( 26) are satisfied for this example. Hence, the system [START_REF] Kundur | Power System Stability and Control[END_REF] has two equilibria and the proof of Theorem 1 implies that the equilibrium with |θ s -φ| > π/2 is locally asymptotically stable. In addition, inequalities (42) of Assumption 2 are satisfied with min = 82.12 and λ = 23.81. Consequently, by Theorem 1, the equilibrium with |θ s -φ| > π/2 is also an almost globally attractive equilibrium. This result coincides with the conclusions in [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF].

The parameters for the second example are taken from [2, Examples 4.2 and 5.1] and are expressed in per unit (pu) including normalized time (for ω Base = 2π60rad/sec), see Table I. The normalized total moment of inertia is given by J = 3 • 2Hω Base , where H is the inertia constant [START_REF] Anderson | Power System Control and Stability[END_REF]. We assume a nominal damping coefficient of 4 . Furthermore, we assume a field current i f = 1.15 pu yielding an RMS amplitude of 1.029 pu for 4 The scaling factor 3 in J and D originates from the following fact. In [START_REF] Anderson | Power System Control and Stability[END_REF], the mechanical equation ( 5) is expressed in pu with respect to the 3phase base power S 3φ . Hence, the pu values of J, D, Tm and Te are also expressed with respect to S 3φ . In the model ( 5), the electrical torque Te in [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF] is expressed with respect to the single-phase power S 3φ

D nom = 3 • 2 pu (see [2, Table D.1-D.5])
3 . Consequently, one way to match our model [START_REF] Andersson | Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic performance[END_REF] with the parameters in [START_REF] Anderson | Power System Control and Stability[END_REF] is to scale J and D by a factor 3, i.e., to represent the mechanical equation ( 5) with respect to the power base

S 3φ 3 .
the EMF e abc , a line resistance of R e = 0.02 pu and line inductance of L e = 0.4 pu.

We start by evaluating local stability of equilibria of the system [START_REF] Kundur | Power System Stability and Control[END_REF]. To this end, we note that condition (26) of Proposition 2 is independent of T m . Furthermore, condition [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF] is satisfied for the chosen set of parameters if the damping coefficient is chosen as D ≥ 6.5D nom . Hence, if D ≥ 6.5D nom , then for any choice of T m such that the inequality ( 13) is strictly satisfied, the system (11) has two equilibria and by Proposition 1 the equilibrium with |θ s -φ| < π/2 is locally asymptotically stable.

Global attractivity can be ensured for the data given in Table I via inequalities (42) for damping factors D ≥ 10D nom . In general, the specific damping factor highly depends on the SG model and, in particular, on how the amortisseur windings are considered in the SG model [START_REF] Anderson | Power System Control and Stability[END_REF]Appendix D]. The SG model [START_REF] Kundur | Power System Stability and Control[END_REF] used in the present paper does not consider explicitly amortisseur damping at all. In that case, it is reasonable to assume a higher damping factor of up to 3•25 pu [2, Appendix D], which clearly implies D ≥ 10D nom .

To further illustrate our findings, we have evaluated the required damping factor D for inequalities (42) to be satisfied for a wide range of operating conditions for both examples discussed above. The results are shown in Fig. 2a for the example from [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF] and in Fig. 2b for the example from [2, Examples 4.2 and 5.1]. In the numerical experiments, we have varied the rotor current i f and the mechanical torque T m , while keeping all other parameters constant. Note that varying T m implies varying the steady-state machine loading. Therefore, we have chosen to plot in Fig. 2 the ratio of required damping D over the nominal damping D nom versus the steadystate RMS amplitude E s = 1/3bω s of the EMF e abc defined in [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF] and the steady-state electrical torque T s e defined in [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF]. All values are in pu and T e is normalized with respect to the single-phase power base given in [START_REF]A method for proving the global stability of a synchronous generator connected to an infinite bus[END_REF], respectively [2, Examples 4.2 and 5.1], i.e., the rated torque corresponds to T s e = 3.0 pu. Recall that large SGs can usually only be loaded at their rated power if the amplitude of the EMF e abc is higher than the rated voltage [START_REF] Fitzgerald | Electric Machinery[END_REF]Chapter 5]. This explains the employed range of values for E s and the limited electrical torque T s e for low values of E s in Fig. 2b. The results show that for both examples the higher the EMF E s , i.e., the higher the rotor current i f , and the higher the machine loading, i.e., the higher T s e , the larger D needs to be in order to ensure that inequalities (42) are satisfied. The latter correlation is more distinct in Fig. 2b, than in Fig. 2a. This may be explained by the higher R/X-ratio of the example in Fig. 2a (R/X = 0.11) compared to that of Fig. 2b (R/X = 0.01), see Remark 5. The made observations coincide with the well-known power-angle characteristic of reduced-order SG models, which states that an SG is more likely to become unstable after a change in load the closer it is operated at its generation limit [START_REF] Grainger | Power System Analysis[END_REF], [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF].

VIII. CONCLUSIONS

A complete analysis of existence and global attractivity of equilibria of a realistic SMIB model has been presented. More precisely, it is shown that (13)-with P defined in (12)-is a necessary and sufficient condition for existence of equilibria. Then, it is proven that if the inequalities (42) of Assumption 2 hold then almost all trajectories converge to a stable equilibrium point. Finally, the conservativeness of the estimates has been assessed via extensive numerical evaluations based on two benchmark problems.

We believe the presented results constitute a promising step towards the development of transient stability analysis methods for power system models with time-dependent transmission lines and physically more realistic SG models. Naturally, the main topic of our future research is the extension of the presented results to the multi-machine case. Given the "scalable" nature of the analysis tools employed here this seems a feasible-albeit difficult-task.

  the mutual inductance. Moreover, in the SMIB model the inductance and resistance of the transmission line connecting the SG to the infinite bus are included in the parameters L and R, see [2, Section 4.13.1]. With these considerations, the electrical equations describing the dynamics of the three-phase stator current i abc : R ≥0 → R 3 are given by

Fig. 1 :

 1 Fig. 1: Schematic representation of a two-pole round-rotor SG based on [32, Figure 3.2]. The abc-axes correspond to the axes of the stator coils denoted by a1,2, b1,2 and c1,2. The axis of coil a is chosen at θ = δ-ϕ = 0. The rotor current i f flows through the rotor windings.The dq-axes denote the rotating axes of the dq-frame corresponding to the mapping T dq (ϕ) with ϕ given in[START_REF] Rüdenberg | Elektrische Schaltvorgänge und Verwandte Störungserscheinungen in Starkstromanlagen[END_REF] and T dq (•) given in Appendix A.

Fig. 2 :

 2 Fig. 2: Evaluation of the required damping factor D relative to the nominal damping factor Dnom for inequalities (42) to be satisfied for a wide range of operating conditions for both investigated examples.

TABLE I :

 I Parameters of the system[START_REF] Kundur | Power System Stability and Control[END_REF] taken from[START_REF] Anderson | Power System Control and Stability[END_REF] Example

	4.4]
	Parameter Numerical value Parameter Numerical value
		in pu			in pu
	ω s	1		J		3 • 1786.94
	Dnom	3 • 2	M f		1.2656
	R	0.0211	L		2.1
	i f	1.15	V		1
	2	1.5	1	0	1	2	3

Almost GAS means that for all initial conditions, except a set of Lebesgue measure zero, the trajectories will converge to the equilibrium.

Please see Section VII for a discussion of the physical meaning of the coefficient D.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous reviewers for many helpful comments that improved the quality and presentation of the manuscript.

APPENDIX

A. The dq-transformation

Following [START_REF] Anderson | Power System Control and Stability[END_REF], [START_REF] Machowski | Power System Dynamics: Stability and Control[END_REF], [START_REF] Teodorescu | Grid Converters for Photovoltaic and Wind Power Systems[END_REF], [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF], the dq-transformation employed in the model derivation of the SMIB model in Section II is stated. Let x ∈ R 3 and ∈ R. Consider the matrix

is called dq-transformation.