Almost global asymptotic stability of a grid-connected synchronous generator

Abstract

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an "infinite bus", i.e. a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo 2π). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the exact swing equation, which resembles a forced pendulum equation and is equivalent to our fourth order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500kW synchronverter.

Stability of the integral control of stable nonlinear systems

“…Of course, the price for choosing a very large T is that we may have to choose a very small gain k, and this may deteriorate the dynamic response of the closed-loop system. Remark 4.3: Similar comments apply if P is almost globally asymptotically stable (aGAS), as defined in [8], [9]. In the latter case, there may be a family Ξ j (u 0 ) ( j ∈ Z) of stable equilibrium points of P corresponding to each constant input u 0 ∈ [u min , u max ], and our main results apply to each such family.…”

“…This theory has been developed with a very specific example in mind: the control of the virtual field current in a synchronverter. Explaining the context of that application would take several pages and instead we just refer to the papers [9], [10]. The material in this paper was originally meant to be a lemma in [10], but then it grew too long.…”

Adaptive Optimizing Control for Nonlinear Synchronous Generator System with Uncertain Disturbance

“…At present, the energy development in the world is mostly hydraulic power generation, thermal power generation, wind power generation and so on. In modern power industry, synchronous generator is the most commonly used alternator, which is widely used in thermal power generation, diesel engine power generation, nuclear power generation, and so on [1][2][3]. During the operation of synchronous generators, there are many system uncertainties and serious external disturbances.…”

Synchronverters used for damping inter-area oscillations in two-area power systems

“…Synchronverters have been introduced precisely to rectify this destabilizing effect of inertia-less inverters working under a variety of control algorithms. A synchronverter is an inverter controlled to emulate the behavior of a synchronous generator, see [27], [25], [26], [14]. The related concept of virtual synchronous generator was introduced earlier in [8].…”

“…A mathematical investigation for the simplest possible case (two identical generators and a load) is contained in [23]. It uses earlier results about the stability of a SG connected to an infinite bus [14]. The aim of this paper is to investigate the effects of virtual friction on some realistic grid models.…”