System modeling and stability analysis is one of the most important issues of inverter-dominated microgrids. It is useful to determine the system stability and optimize the control parameters. The complete small signal models for the inverter-dominated microgrids have been developed, which are very accurate and could be found in literature. However, the modeling procedure will become very complex when the number of inverters in microgrid is large. One possible solution is to use the reduced-order small signal models for the inverter-dominated microgrids. Unfortunately, the reduced-order small signal models fail to predict the system instabilities. In order to solve the problem, a new modeling approach for inverter-dominated microgrids by using dynamic phasors is presented in this paper. Our findings indicate that the proposed dynamic phasor model is able to predict accurately the stability margins of the system, while the conventional reduced-order small signal model fails. In addition, the virtual ${\boldsymbol{\omega }}$ -E frame power control method, which deals with the power coupling caused by the line impedance X/R characteristic, has also been chosen as an application example of the proposed modeling technique.

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Dynamic Phasors-Based Modeling and Stability Analysis of Droop-Controlled Inverters for Microgrid Applications

The characteristics of the networks which are composed of many randomly connected threshold elements are investigated with the intention of understanding some aspects of information processing in nervous systems. In these networks, the statistical properties of connection are sufficient to determine the characteristics. Information is carried by the activity level of a network which designates the rate of exciting elements. Dynamics of the activity level is studied. Two statistical parameters, which are sufficient to determine the characteristics of networks, are extracted and the networks are categorized into three classes by these parameters. One is monostable, having only one stable activity level. Another is monostable or bistable according to the average threshold value of the elements. The third is astable or monostable. The characteristics of these three kinds of networks are analyzed in detail. Various systems can be obtained by connecting random networks, where the average thresholds of component networks can be controlled by other networks. The system performances are given. A stable oscillation of a long period is shown to exist in a system composed of two kinds of elements, i.e., excitatory and inhibitory elements, by randomly connecting them. A model for association of ideas is presented.

Characteristics of randomly connected threshold-element networks and network systems

Describing the small signal performance of a multi- machine synchronous power system by a set of differential equations of the form [x] = [A] [x] allows standard multivariable control theory to be used in dynamic stability studies. The construction of the [A] matrix for a multimachine power system involves the application of Kron's rotational transformation to the transmission network admittance matrix, and a matrix analysis of the synchronous machines using internal flux linkages as state variables.

Dynamic Stability Calculations for an Arbitrary Number of Interconnected Synchronous Machines

The selection of stabilizing signals to improve power system dynamic behavior is presented. Some results from modern control theory are used to show how such stabilizing signals may be derived systematically for a turboalternator model connected through a double circuit line to an infinite bus. The closed-loop performance of this nonlinear system model with the derived linear optimal controller is then evaluated for a wide range of input disturbances and operating conditions which include three-phase fault studies with autoreclosure.

The control of a synchronous machine using optimal control theory

This paper deals with recent advances in developing direct methods for studying the transient stability problem of single-machine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Computer simulation methods are then briefly discussed, and it is indicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov's second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed.

Direct method for transient stability studies in power system analysis

The design of synchronous-generator automatic regulating equipment by linearisation techniques is now firmly established. The paper presents an obvious extension of one method, where, by means of matrix algebra, a set of general coefficients related to the well known Heffron and Phillips constants is derived, which allows complex impedance terms to be included in the analysis. In keeping with modern practice, these constants are incorporated in equations which are formulated as sets of first-order differential equations. This form provides a basis for future applications of optimal control theory, and, in addition, may be scaled more easily in both time and amplitude for analogue-simulation studies. A detailed example of a multiloop a.v.r. connected to a 30MW machine is given for reference purposes. The matrix methods, used in conjunction with the matrix iterative techniques of load-flow analysis, extend naturally to multimachine systems, allowing the dynamic operating and stability problem to be investigated with respect to possible future multimachine co-ordinated control schemes.

M.A. Laughton

Papers

Matrix analysis of dynamic stability in synchronous multimachine systems