Comparison of Coupled Nonlinear Oscillator Models for the Transient Response of Power Generating Stations Connected to Low Inertia Systems

Abstract

Coupled nonlinear oscillators, e.g., Kuramoto models, are commonly used to analyze electrical power systems. The cage model from statistical mechanics has also been used to describe the dynamics of synchronously connected generation stations. Whereas the Kuramoto model is good for describing high inertia grid systems, the cage one allows both high and low inertia grids to be modelled. This is illustrated by comparing both the synchronization time and relaxation towards synchronization of each model by treating their equations of motion in a common framework rooted in the dynamics of many coupled phase oscillators. A solution of these equations via matrix continued fractions is implemented rendering the characteristic relaxation times of a grid-generator system over a wide range of inertia and damping. Following an abrupt change in the dynamical system, the power output and both generator and grid frequencies all exhibit damped oscillations now depending on the (finite) grid inertia. In practical applications, it appears that for a small inertia system the cage model is preferable.Index Terms-Power system stability, Power system transients, Power system protection, Rate of change of frequency or ROCOF, Renewable energy sources, Synchronous generators.

Active Damping of Power Oscillations Following Frequency Changes in Low Inertia Power Systems

“…To study the cascaded transfer function analytically we select an input signal corresponding in general to an actual signal of the generator during a transient event capable of yielding the stabilizing signal PSS V in closed form. To analyze the linear transient response due to a sudden small change in the maximum electromagnetic torque max el  [12] (see Appendix A) at the instant t = 0, this input is represented as the damped oscillation, The constants 0 a , 0 b , and V  can then be found explicitly as follows. The input to the PSS1A is usually the rotor angular speed deviation [15].…”

“…Here j  represent the eigenvalues of our dynamical system whereas the amplitudes 0 j a are determined from the corresponding eigenfunctions and the initial conditions [12]. As in V  , for any 0 , jj a   an equivalent conjugate term must exist, whereas for any j   , 0 j a  also.…”

“…However, recent measurements cannot be explained by this Ansatz as generator-grid feedback is ignored. Therefore, we recently developed new dynamical methods based on a rotating double pendulum [10], [12]. Our model (v. Fig.1) [10], [12] is two rotating masses representing on the one hand the inertia Jgen of the gridconnected synchronous generating unit and on the other the grid itself represented by the inertia Jgrid.…”

“…Therefore, we recently developed new dynamical methods based on a rotating double pendulum [10], [12]. Our model (v. Fig.1) [10], [12] is two rotating masses representing on the one hand the inertia Jgen of the gridconnected synchronous generating unit and on the other the grid itself represented by the inertia Jgrid. The work will be based on a recent paper [12] where appropriate dynamical equations for low inertia grids (as summarized in Appendix A) were written as differential-recurrence relations so that matrix algebra yields the relevant characteristics (based on methods developed in [13] and [14]).…”

“…Our model (v. Fig.1) [10], [12] is two rotating masses representing on the one hand the inertia Jgen of the gridconnected synchronous generating unit and on the other the grid itself represented by the inertia Jgrid. The work will be based on a recent paper [12] where appropriate dynamical equations for low inertia grids (as summarized in Appendix A) were written as differential-recurrence relations so that matrix algebra yields the relevant characteristics (based on methods developed in [13] and [14]). Specifically, the nonlinear response of the rotor angle to a disturbance is given as a sum of eigenfunctions which is our basis for critically examining the adequacy of the common power system stabilizers type PSS1A and its tuning.…”