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The purpose of this paper is to present the Galerkin method for analysis of steady-state processes in periodically time-varying circuits.

A converter circuit working on a time-varying load is often controlled by different signals. In the case of incommensurable frequencies, one can find a steady-state process only via calculation of a transient process. As the obtained results will not be periodical, one must repeat this procedure to calculate the steady-state process on a different time interval. The proposed methodology is based on the expansion of ordinary differential equations with one time variable into a domain of two independent variables of time. In this case, the steady-state process will be periodical. This process is calculated by the use of the Galerkin method with bases and weight functions in the form of the double Fourier series.

Expansion of differential equations and use of the Galerkin method enable discovery of the steady-state processes in converter circuits. Steady-state processes in the circuits of buck and boost converters are calculated and results are compared with numerical and generalized state-space averaging methods.

The Galerkin method is used to find a steady-state process in a converter circuit with a time-varying load. Processes in such a load depend on two incommensurable signals. The state-space averaging method is generalized for extended differential equations. A balance of active power for extended equations is shown.

The purpose of this study is to identify a novel methodology for direct calculation of steady-state periodic solutions for electrical circuits described by nonlinear differential equations, in the time domain.

An iterative algorithm was created to determine periodic steady-state solutions for circuits with nonlinear elements in a chosen set of time instants.

This study found a novel differential operator for periodic functions and its application in the steady-state analysis.

This approach can be extended to the determination of two- or multi-periodic solutions of nonlinear dynamic systems.

The complexity of the steady-state analysis can be reduced in comparison with the frequency-domain approach.

This study identified novel difference equations for direct steady-state analysis of nonlinear electrical circuits.

Reports on a comparison by computer simulation between conventional periodic loading (PL) and job‐group loading (JGL). When conventional PL is used, the search of the best part input sequence must be performed in order to optimize performance of a flexible manufacturing system (FMS). JGL works, instead, as a dynamic rule for real time scheduling of FMS, defining a part releasing policy able to guarantee the reaching of a periodic steady state without non‐productive times on the bottleneck workstation. However, JGL does not assure, in some cases, the same performance arising from the optimal part input sequence of conventional PL, in terms of non‐productive times in FMS filling and emptying phases, work in progress and throughput time. The paper demonstrates that using any JGL rule or the best PL part input sequence gives rise to negligible differences in FMS performance. Furthermore, the dynamic capabilities of the JGL also allow for spontaneously restoring the FMS periodic steady state without non‐productive times after any transient, for instance when production mix changes occur.

To introduce an efficient methodology for the computation of the periodic steady state solution of power systems with nonlinear and time‐varying components which combines a Newton method based on a numerical differentiation procedure to obtain a fast steady state solution in the time domain and parallel process techniques.

Nonlinear electric systems are represented by a set of differential equations, the conventional solution in the time domain is accelerated by a Newton method based on a numerical differentiation procedure for the convergence of state variables to the limit cycle and thus to the network periodic steady state solution. The efficiency of the solution is further enhanced with the application of parallel processing technology based on parallel virtual machine (PVM) and multi‐threading (MT).

The periodic steady state solution of nonlinear electric systems, even of large‐scale, can be efficiently obtained in the time domain with the application of Newton methods for the fast converge of state variables to the limit cycle. The efficiency of the computer solution can be dramatically enhanced with the application of parallel processing technology. The potential of the PVM and MT platforms is shown in the investigation. A comparison of advantages and disadvantages associated with each parallel processing platforms is given; a quantitative comparison between PVM and MT is provided.

The steady state solution of nonlinear electric systems can be efficiently obtained with a combination of Newton methods for the convergence acceleration to the limit cycle and parallel processing techniques.

The steady state solution of nonlinear electric systems using a Newton method based on a numerical differentiation procedure for the convergence acceleration to the limit cycle and parallel processing based on the PVM and MT platforms has not, to the authors' knowledge, reported before.

This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models.

Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only.

An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain.

Reduction of software and hardware requirements for determining steady-states of electromagnetic.

A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form.

Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions.

The development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions.

Novel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic.

Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist.

Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time.

Identify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

The purpose of this study is to determine the steady state of an electromagnetic structure using the finite element method (FEM) without calculation of the transient state. The proposed method permits to reduce the computation time if the transient state is important.

In the case of coupling magnetic and electric circuit equations to obtain the steady state with periodic conditions, an approach can be to discretise the time with periodic conditions and to solve the equation system. Unfortunately, the computation time can be prohibitive. In this paper, the authors proposed to use the waveform relaxation method associated with the Newton method to accelerate the convergence.

The obtained results show that the proposed approach is efficient if the transient state is important. On the contrary, if the transient state is very low, it is preferable to use the classical approach, namely, the time-stepping FEM.

The main limitation of the proposed approach is the necessity to evaluate or to know the time constant and consequently the duration of the transient state. Moreover the method requires some important memory resources.

In the context of the use of the time-stepping FEM, one of the problems is the computation time which can be important to obtain the steady state. The proposed method permits avoidance of this difficulty and directly gives the steady state.

The proposed approach will permit to model and study the electromagnetic systems in the steady state, and particularly the transformers. Because of the gain in computing time, the use of optimisation techniques will be facilitated.

The novelty of this study is the proposal of the waveform relaxation–Newton method to directly obtain the steady state when applied to the three-phase transformer.

The purpose of this study is to present the performance evaluation of three shooting methods typically applied to obtain the periodic steady state of electric power systems, with the aim to check the benefits of the use of cloud computing regarding relative efficiency and computation time.

The mathematical formulation of the methods is presented, and their parallelization potential is explained. Two case studies are addressed, and the solution is computed with the shooting methods using multiple computer cores through cloud computing.

The results obtained show a reduction in the computation time and increase in the relative efficiency by the application of these methods with parallel cloud computing, in the problem of obtainment of the periodic steady state of electric power systems in an efficient way. Additionally, the characteristics of the methods, when parallel cloud computing is used, are shown and comparisons among them are presented.

The main advantage of employment of parallel cloud computing is a significant reduction of the computation time in the solution of the problem of a heavy computational load caused by the application of the shooting methods.

The purpose of this paper is to introduce a method for the analysis of steady-state processes in periodically time varying circuits. The method is based on a new definition of frequency responses for periodic time-varying circuits.

Processes in inverter circuits are often described by differential equations with periodically variable coefficients and forcing functions. To obtain a steady-state periodic solution, the expansion of differential equations into a domain of two independent variables of time is made. To obtain differential equations with constant coefficients the Lyapunov transformation is applied. The two-dimensional Laplace transform is used to find a steady-state solution. The steady-state solution is obtained in the form of the double Fourier series. The transfer function and frequency responses for the inverter circuit are introduced.

A set of frequency characteristics are defined. An example of a boost inverter is considered, and a set of frequency responses for voltage and current are presented. These responses show a resonance that is missed if the averaged state-space method is used.

A new definition of frequency responses is presented. On the basis of frequency responses, a modulation strategy and filters can be chosen to improve currents and voltages.

FMS real time scheduling requires the concurrent solution of both loading and despatching problems. The loading strategy is the most critical and important scheduling decision in an FMS. Presents an effective loading rule (job‐group loading rule), based on experimental observations: a group of jobs, reflecting the ratios of the production mix, is loaded on the input warehouse every time that it is completely empty; all the jobs loaded on the input warehouse have to be accessible for the material handling system. Demonstrates how this job‐group loading works as a dynamic rule for real time scheduling of manufacturing systems, defining a part releasing policy able to guarantee reaching a periodic, steady state production. The capability of the job‐group loading rule has been verified when the FMS characteristics are closest to a real situation. Analyses the effects of the interferences on transport operations, the variability of machining times, the finite capacity of interoperational storage and the stops for preventive maintenance or breakdown. Research results show the capability in FMS management of real time scheduling based on the job‐group loading rule.