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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.

EDMC represents extended dynamic matrix control, which can be applied to nonlinear process control. In this method, control inputs are determined based on a linear model that approximates the process and is updated during each sampling interval. Since nonlinear relation still exists between the prediction error and the control input, numerical (iterative) methods are used to solve the optimization problem defined in the method. For nonlinear processes with high variation and/or sign changes in their steady‐state gain, iterative methods do not converge properly to an acceptable solution for some desired outputs or external disturbances. To eliminate the problem, we augment the process with its steady‐state gain inverse (or pseudo inverse whenever required) such that the steady‐state gain for the new augmented system is constant or contains slow variations. In the case of unstable processes, the method may be applied after stabilizing the process using a proper state or output feedback. Effectiveness of the method has been examined using computer simulations of some benchmark processes. Some of the obtained results are presented in this paper.

The purpose of this paper is to propose a new method for mathematical modeling of the buck dc‐dc converter in discontinuous conduction mode (DCM). By using the presented modeling method, the analysis of the transient and the steady states of the buck dc‐dc converter can be performed.

The proposed method is based on the two Laplace and Z transforms. In the proposed method, at first, the equations of the inductor current and the capacitor voltage are obtained as the power switch is on and off. Then by using the Laplace and Z transforms, the obtained equations are solved and the relations of the inductor current and the output voltage are obtained. In the proposed method, the Laplace transform is used for determining of the general relations of the inductor current and the output voltage. Also the Z‐transform is used as a tool for determining the initial values of the inductor current and the output voltage.

The transient and the steady state response of the dc‐dc converter is analyzed by the proposed method. By using the Z‐transform, the transient response of the converter and the effect of the elements of the converter on the time constant of the transient response are investigated. In addition, the effect of the elements of the converter and the load resistance on the electrical parameters of the converter such as the output voltage ripple and the inductor current ripple are investigated.

The proposed method in this paper is a suitable method for mathematical modeling of dc‐dc converters. The acernote of this method is that it can be used in both transient and steady state response, analysis of the dc‐dc converters. By using the final value theorem of the Z‐transform, the steady state response of the converter is investigated. Also by using this transform, the time constants of the transient response of the converter are determined. Finally, the results of the theoretical analysis are compared with the results of simulation in PSCAD/EMTDC and also the experimental results to prove the validity of the presented subjects.

The purpose of this paper is to introduce methods for calculating steady‐state and transient processes in a symmetrical three‐phase matrix‐reactance frequency converter (MRFC). The MRFC in question makes it possible to obtain a load output voltage much greater than the input voltage.

MRFCs based on a matrix‐reactance chopper are used for both frequency and voltage transformation. The processes in a MRFC system are described by nonstationary differential equations. A two‐frequency complex function method is proposed for solving non‐stationary equations in steady‐state. The method is applied to a state‐space averaged mathematical model used in the analysis of the discussed MRFC. A two‐frequency matrix transform is proposed for solving non‐stationary equations. This method can be used to find both transient and steady‐state processes.

The two‐frequency complex function method permits the reduction from 12 non‐stationary differential equations to four stationary differential equations. The two‐frequency matrix transform allows the transformation of non‐stationary differential equations to stationary ones. By using these methods descriptions of steady‐state and transient properties of buck‐boost MRFCs are obtained.

A new method of solving of nonstationary differential equations is presented. The method is useful for process analyses in nonstationary power electronic converters.

The purpose of this paper is to reduce issues arising when computing steady‐state solutions for AC machine models using the harmonic balance method.

Generally, currents at steady‐states of AC machines are described by periodic or quasi‐periodic time functions, which Fourier spectra are determined by an infinite set of algebraic equations obtained from a harmonic balance method. To solve them, after reducing to finite dimensions, an iterative algorithm is developed in this paper. It bases on the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine. Since that decomposition is done separately, due to a band type form of this matrix, the equation set determining the Fourier spectra of currents is solved recurrently.

An algorithm for the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine and an iterative algorithm for determining AC machine steady‐state currents in a recursive manner.

The approach is limited to solving of so‐called “circuital” models of AC voltage supplied machines. The approach breaks the large dimension barrier when solving steady‐state equations for AC machines.

Reducing computer requirements in terms of computer memory, workload and computing time to determine a steady‐state solution for AC machines.

A separation of the LU decomposition of an infinite matrix representing the inductance matrix in AC machine steady‐state model from the solution method.

This paper aims to present two methods for calculating the steady state probability of a repairable fault tree with priority AND gates and repeated basic events when the minimal cut sets are given.

The authors consider a situation that the occurrence of an operational demand and its disappearance occur alternately. We assume that both the occurrence and the restoration of the basic event are statistically independent and exponentially distributed. Here, restoration means the disappearance of the occurring event as a result of a restoration action. First, we obtain the steady state probability of an output event of a single‐priority AND gate by Markov analysis. Then, we propose two methods of obtaining the top event probability based on an Inclusion‐Exclusion method and by considering the sum of disjoint probabilities.

The closed form expression of steady state probability of a priority AND gate is derived. The proposed methods for obtaining the top event probability are compared numerically with conventional Markov analysis and Monte Carlo simulation to verify the effectiveness. The result shows the effectiveness of the authors’ methods.

The methodology presented shows a new solution for calculating the top event probability of repairable dynamic fault trees.

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.

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.

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.