On the stability of equilibrium for a reformulated foreign trade model of three countries

In this paper, we study the stability of equilibrium for a foreign trade model consisting of three countries. As the gravity equation has been proven an excellent tool of analysis and adequately stable over time and space all over the world, we further enhance the problem to three masses. We use the basic Structure of Heckscher-Ohlin-Samuelson model. The national income equals consumption outlays plus investment plus exports minus imports. The proposed reformulation of the problem focus on two basic concepts: (1) the delay inherited in our economic variables and (2) the interaction effect along the three economies involved. Stability and stabilizability conditions are investigated while numerical examples provide further insight and better understanding. Finally, a generalization of the gravity equation is somehow obtained for the model

A macroeconomic mathematical model for the national income of a union of countries with interaction and trade

In this article, we assume a union of countries where each national economy interacts with the others. We propose a new model where (a) delayed variables are incorporated into the system of equations and (b) the interaction element is restricted into the annual governmental expenditure that is determined according to the experience of the total system and the trade relations of these countries (exports-imports). In addition, we consider the equilibrium(s) of the model (a discrete-time system) and study properties for stability, the appropriate control actions as well as the total system design in order to obtain a stable situation. Finally, a practical application is also investigated that provides further insight and better understanding as regards the system design and produced business cycles

Modal Participation Factors of Algebraic Variables

This paper proposes an approach to determine the participation of algebraic variables in power system modes. The approach is based on a new interpretation of the classical participation factors, as well as on the definition of adequate output variables of the system's state-space representation. The paper considers both the linear and generalized eigenvalue problems for the calculation of the participation factors and presents a theorem to cope with eigenvalue multiplicities. An illustrative example on the two-area system, as well as a study on a 1479-bus dynamic model of the all-Irish transmission system are carried out to support the theory and illustrate the features of the proposed approach.Science Foundation Irelan

Generalized fractional controller for singular systems of differential equations

In this article we consider a class of singular linear systems of first order, and introduce a generalized fractional order feedback controller of Caputo type. The closed loop system in question is a singular system of differential equations having both first, and fractional order derivatives. We provide a comprehensive theory for the existence and uniqueness of solutions, as well as for the stability of the system with inclusion of the fractional order controller. An example of a singular system with a fractional order proportional integral controller, as well as an example on a 3-bus power system with inclusion of a fractional order damping controller, is given to illustrate our theory.Science Foundation Irelan

The Möbius transform effect in singular systems of differential equations

The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of differential equations and the solutions of its proper M-systems, i.e., systems that appear after applying the generalized Möbius transform to the pencil of the original singular system (prime system). Firstly, we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of its proper M-systems. Moreover these solutions can be explicitly represented without resorting to any further processes of computations. Finally, numerical examples are given to illustrate our theory.Science Foundation Irelan

Frequency Divider as a Continuum

The letter describes a novel, continuum-based approach, to capture the evolution of electromechanical dynamics in a power network following a disturbance. Such approach is based on the frequency divider formula (FDF), which was recently proposed by the third author. A key point in obtaining the frequency divider formula (FDF) as a consequence of a continuum, is to show that the spatial rate of change, at a given time, of the frequency along a lossless line is constant. The proposed derivation is then compared with the electromechanical wave approach (EWA), which has been discussed in the literature in a variety of hues and is aimed at modeling the propagation in a power system of frequency oscillations following a disturbance. The discussion illustrates similarities and differences between FDF and electromechanical wave approach (EWA).European Commission Horizon 2020European CommissionSchweizerischer Nationalfonds zur Förderung der Wissenschaftlichen ForschungNCCR AutomationSustainable Energy Authority of IrelandFRESLIPSEdgeFLE

Applications of the Frenet Frame to Electric Circuits

The paper discusses the relationships between electrical quantities, such as voltages, currents, and frequency, and geometrical ones, namely curvature and torsion. The proposed approach is based on the Frenet frame utilized in differential geometry and provides a general framework for the definition of the time derivative of electrical quantities in stationary as well as transient conditions. As a byproduct, the proposed approach unifies and generalizes the time- and phasor-domain frameworks. Other noteworthy results are a new interpretation of the link between frequency and the time derivatives of voltage and current; and a definition of the rate of change of frequency that includes the novel concept of 'torsional frequency.' Several numerical examples based on balanced, unbalanced, harmonically-distorted and transient voltages illustrate the findings of the paper.Science Foundation IrelandEuropean Commission Horizon 202

Damage Accumulation in Cyclically-Loaded Glass-Ceramic Matrix Composites Monitored by Acoustic Emission

Barium osumilite (BMAS) ceramic matrix composites reinforced with SiC-Tyranno fibers are tested in a cyclic loading protocol. Broadband acoustic emission (AE) sensors are used for monitoring the occurrence of different possible damage mechanisms. Improved use of AE indices is proposed by excluding low-severity signals based on waveform parameters, rather than only threshold criteria. The application of such improvements enhances the accuracy of the indices as accumulated damage descriptors. RA-value, duration, and signal energy follow the extension cycles indicating moments of maximum or minimum strain, while the frequency content of the AE signals proves very sensitive to the pull-out mechanism